Applying for PhDs with no research experience?

Personally, my only research experience came from my undergrad and masters dissertations, I hadn't completed any summer internships or anything. But you will likely be at a disadvantage if you get an interview as you won't have any experiences to draw on. I'm surprised that as a STEM student you won't have had to do a research project at undergrad or masters level.

I would suggest trying to get some research experience, not only for your PhD application, but to see whether you actually want to do research or not. Working in research and doing a PhD can be very different to how people perceive it and taking on a PhD is a big commitment and not one to take lightly.

Your dissertation(s) would be the best option for demonstrating research experience. Also just contact lecturers who are doing research in an area of interest and ask if there is any opportunity to do research with them over this summer (albeit not necessarily funded). Your dissertation supervisor would be a good place to start. I do know that sometimes supervisors will arrange for a summer project after final year to continue with work from your dissertation that you did with them, which likely wouldn't get arranged until after deadlines for formally funded summer research opportunities. Likewise you may be able to do the same after your masters course.

Unfortunately I had the option to do an undergrad project this academic year as an optional module but I opted to do a theoretical module instead (without thinking it would have been a wise choice).

Of course there will be a summer project component in the masters I’m going to do, but that I will be starting the summer after making a PhD application. The only options I can think of would be to contact an academic personally to see if they could offer a project for me to do, or to just do a project off my own back with no supervision. I’m not sure if these would be credible alternatives though (especially the latter).