So I already asked two of my core teachers for letters of recommendation to the colleges I will be applying to, but I’ve been working really closely with one of my elective teachers for an independent study. Can this teacher submit a supplement letter? Do colleges allow an extra letter of recommendation?!?!
Yes, most colleges allow an extra letter of recommendation. I would take caution in deciding whether to let this additional teacher write a letter, however, because colleges severely discourage additional letters that aren’t extraordinarily revealing about some otherwise unknown aspect of your character. If this elective teacher can vouch for a different side of you than your other two teachers, then perhaps you should ask him/her to write the letter.
Plus, most colleges want zero rec letters. A few will take one. A smaller amount want two. The ones that accept three are very few
@T26E4 are you referring to additional (supplemental) rec letters or all rec letters?
All rec letters. The vast vast majority of colleges require zero rec letters. I recall sitting at a Purdue engineering info session. The admissions officers shared how they begrudgingly started taking rec letters and essays b/c their peer schools were doing it. They had no problem filling their classes with excellent students and their engineering prowess is quite hard to beat.
That’s really neat, I didn’t think of it that way. 
While what @T26E4 says is true, there are still some schools that require recs. Whether they accept additional recs varies by the college, so check the websites. Having said that, I’m not really sure what a 3rd teacher rec will tell the AO that the other 2 didn’t already cover. IMO, an extra rec from a coach, EC moderator, or boss would add more than another teacher rec. There’s an old saying - “The thicker the file, the thicker the kid.” Be careful.
Many schools do require recommendation letters so you are wise to be on top of getting that handled. Just be sure that your supplemental recommendation (which is not required at all) can speak to non-classroom related aspects of you and your work.
Thank you @bzhang42 , @T26E4 , @DeeperBlue25 , @skieurope , and @happy1 for your responses! I really appreciate it. I think my teacher can definitely speak for other qualities apart from my core teachers. If some colleges I apply to don’t accept more than 1 or 2 letters, then I will just leave out the supplemental.