Not all suppliers run a credit check, but those that do will leave a record of their search on your credit file. Soft searches cannot be seen by others, so they don't affect your credit rating at all. Hard searches only remain for up to 12 months before they are removed. So it is not there permanently. A hard search won't necessarily have a negative affect on your credit rating, unless you have racked up loads of them in quick succession. That just makes you look desperate, and lenders do not like that one bit.
So it won't hurt to try and see whether Vodafone will accept you. Typically, phone and/or broadband contracts are not as difficult to get as, say a credit card or a loan, where the risk to the lender is far greater. So long as you pass their affordability checks and are not moving house too often (they like stability), they are less bothered by employment status than other financial products which have higher monthly payments. Where you are more likely to run into problems is if you have a thin credit file and have never had any credit products before. That's another reason to try and get a broadband contract on there to start building up a solid credit history for when you want to borrow money later on in life.
Everybody needs to start somewhere.