Can I study vet nursing with a BTEC applied science course?

Will I be able to go into vet nursing with the BTEC applied science extended certificate? I wasn’t able to get a 6 in math so I couldn’t study the a level sciences. I plan on retaking it this summer to get a 5 or 6 to improve my chances. I currently study a level history and sociology alongside it, but will have to drop one to study the diploma applied science which causes a problem since I currently really enjoy both of these and got a 9 in history gcse. If not can I still apply to a foundation year with just the extended certificate?

You're a tad inconsistent with the size of BTEC in Applied Science you're referring to.
First you say "the BTEC applied science extended certificate", then "the diploma applied science", and then back to "the extended certificate" again.
Pearson (who own the BTEC brand) offer the following BTECs in Applied Science:
• Pearson BTEC Level 3 National Certificate in Applied Science (worth 0.5 A levels)
• Pearson BTEC Level 3 National Extended Certificate in Applied Science (worth 1 A level)
• Pearson BTEC Level 3 National Foundation Diploma in Applied Science (worth 1.5 A levels)
• Pearson BTEC Level 3 National Diploma in Applied Science (worth 2 A levels)
• Pearson BTEC Level 3 National Extended Diploma in Applied Science (worth 3 A levels)
Which size of BTEC are you referring to?

Just for clarity you're asking whether the Pearson BTEC Level 3 National Extended Certificate in Applied Science (worth 1 A level) would suffice as you don't want to do the Pearson BTEC Level 3 National Diploma in Applied Science (worth 2 A levels), as that would mean dropping one of your existing A levels. Have I got that right?
(Edited as I originally misunderstood the question.)

Yes! That’s correct. I’m asking whether I would still be able to pursue vet nursing without doing the national diploma level since I’d have to drop one of my current a levels.

Thanks for confirming. There aren't too many providers of straight Veterinary Nursing degrees in the UK - which helps in terms of establishing which qualifications are acceptable and which aren't.
• Hartpury University want "MMM-DMM in an Extended Diploma in a relevant subject", although they do say, "We will consider combinations of level three qualifications." You'd have to contact them to establish whether your particular combination would be acceptable or not, and at what grades. (Details)
• Middlesex University want "DMM-DDM in a related subject" (Details) However, they also say here that "Where you are studying a combination of qualifications (e.g. A level and BTEC), we provide the UCAS tariff points we require." So they might accept you combination of qualifications, especially as at A level they only need one science - which an Extended Certificate in Applied Science surely is
• University Centre Reaseheath want "Pearson BTEC Level 3 National Extended Diploma (first teaching from September 2016) – A minimum of grade DMM in Applied Science or Animal Care/Management" or "Pearson BTEC Level 3 National Diploma (first teaching from September 2016) – a minimum of grade D*D* in Applied Science or Animal Care/Management". They also say they they will accept, "A minimum of 112 tariff points from A & AS levels, including at a least a grade C in one of the following subjects: GCE A Level: Biology, Chemistry, Science or Applied Science". That would tend to suggest that your an Extended Certificate in Applied Science (plus your two A levels) would be acceptable. (Details)
• Royal Veterinary College, University of London want "BTEC Level 3 National Extended Diploma in Applied Science / Applied Science (Biomedical Science) DDM overall (no specific unit requirements)." They make no mention of BTECs smaller that the Extended Diploma. (Details)
• SRUC Scotland's Rural College don't mention BTECs in their entry requirements. (Details)
• Warwickshire College and University Centre want "at least 104 UCAS points which may be derived from a Level 3 qualification in a related subject or A Levels including one Science subject, ideally Biology." The fact that that say they want the points to have come from a (singular) Level 3 qualification of A levels leads me to believe that they're looking for the Extended Diploma. (Details)
• Ulster University want "Pearson BTEC Level 3 Extended Diploma in Applied Science (RQF) – Award profile DMM" or "Pearson BTEC Level 3 National Extended Certificate in Applied Human Biology - Award profile of M plus A Level Grades BC" or "Pearson BTEC Level 3 National Extended Diploma in Animal Management (RQF) (1080 GLH) 603/3043/0 with an award profile of DMM". Your combination would therefore not be acceptable. (Details)
There are also some which offer a "with Foundation Year" option, like Middlesex University, but they're just as vague about the qualifications they'd accept.
The other option is to take a Foundation Degree (FdSc), and then a "top-up" course to get the full degree. Is that something you've considered?
Or, if none of the above appeal, a few places offer a Veterinary Nursing and Companion Animal Behaviour degree (or similar). You should be able to get a good overview of what is available in terms of Veterinary Nursing generally by using this link to UCAS Search.

Thank you so much for all the detail! I’m not sure wha a top up course is? Would you recommend me emailing some of the uni’s above to confirm before dropping an a level to do the extended diploma?

The top-up course would only apply if you did a Foundation Degree (FdSc) rather than a full Bachelor's degree (BSc). It's actually easier to explain if we talk in terms of levels. The term "level" essentially describes how advanced the course is.
Level 3 means things like A levels, T levels, BTECs, and all the normal qualifications you might do immediately after GCSEs.
Level 4 is the first teaching year of a standard 3-year undergraduate degree.
Level 5 is the second teaching year of a standard 3-year undergraduate degree.
Level 6 is the third teaching year of a standard 3-year undergraduate degree.
Level 7 and above are things like Master's degrees and PhDs.
(I've used the phrase "teaching year" as many courses these days, including Veterinary Nursing courses, include a non-teaching placement year.)
So once you've done your a standard three teaching year degree, you have a Level 6 qualification - an honours degree like BSc (Hons) or BA (Hons). Another option would be to take what's called a Foundation Degree (FdSc), which is a Level 5 qualification. You could then top it up to a full degree by taking a one-year top-up course. Don't let the word "Foundation" confuse you here - this is very different from the foundation year which some degree courses add to the start of the degree to prepare you for it (a foundation year is actually at Level 3).
The only reason I mentioned a Foundation Degree is that I saw that a lot of universities offer it for Veterinary Nursing. I don't know why. For example:
• Foundation Degree in Science Veterinary Nursing at the Royal Veterinary College (Details)
• Veterinary Nursing FdSc at Nottingham Trent University (Details)
• FDSc Veterinary Nursing at Aberystwyth University (Details)
And yes, if it's not clear from a uni's web site that they would accept a BTEC Level 3 Extended Certificate in Applied Science plus A levels in History and Sociology, you're going to need to contact them and ask.

Thank you so much for all your help! I’ll have to research more into this. Thank you