I have had a chemical engineering internship for a year so far and would like to continue research in college. I am thinking of applying to the University of California, San Diego, but I have read that with the large population and so many competitive STEM students, resources can run short and it is difficult to get the class you want. Does that apply to internships in UCSD as well?
If you already obtained an internship by yourself, the school you attend won’t hinder you from getting the next one. 
Based on a good friend’s experience, the UCSD career center will have jobs/internships posted but you are mostly on your own to find internships. As stated above, you managed to get one internship so far, so you should not have any problems finding another.