Can someone with Social Anxiety be discriminated against when applying for college or a job; and what can I do to limit this?

They'll only know if you tell them. If it is obvious in your behavior and your behavior negatively impacts the hiring process, then they'll chose against you because they need a different behavior. I'm an introvert and would MUCH MUCH rather not talk with people. Most of my day is spent quietly behind a computer, studying data and working through spreadsheets while I listen to music on headphones. But I work in marketing, which is a very social field. I found a way to stay true to my personality and what works for me while working in a field that doesn't typically celebrate that personality. Maybe in the vet world you can be a surgical support member, or a surgeon, or something that doesn't require interacting with clients. That will require you to be extremely technically competent. I've hired people in the past who had lacked social skills but were so technically competent that their technical skills more than made up for their lack of social skills.

Lastly, be careful how you apply a label to yourself. To repeat your phrasing, you HAVE social anxiety. It is ONE thing that you have. Much like you have hair, hands, a sense of humor, and all of the other physical and psychological elements that contribute to who you are. Social anxiety is just one thing that you have. It's not the entirety of who you are. We each care way way way more about our weaknesses and traits than the larger world cares. The truth of the matter is that the world hardly ever pays attention to us. Which cuts both ways. It ignores you and your weaknesses most of the time, but it also ignores your successes and strengths most of the time.

You can do anything you put your mind to. I completely understand, my daughter has social anxiety as well . She has also had a hard time finding employment due to the quiet demeanor . She was advised to go to work in a library. I think that if you go to school to be what you want to be you will be great. I would volunteer at an animal shelter or try to go to small activities that are outside so you do not feel overwhelmed. Start by collecting supplies to take to the shelter or signing up online . A lot of job opportunities have come my way because I have volunteered. Keep up the great work , you can do it!
https://www.paws.org/get-involved/volunteer/
https://www.animalhumanesociety.org/volunteer/volunteer-opportunities

Companies are realizing it is important to have diversity in their teams. Gone are the days that being extroverted is the only way to be a team player. Susan Cain has a wonderful book that I would suggest reading "Quiet". You will find that there are a lot of people who are great contributors to their company and they too, don't enjoy all of the social interaction. Find your strength and use that to your advantage. https://en.wikipedia.org/wiki/Quiet:_The_Power_of_Introverts_in_a_World_That_Can%27t_Stop_Talking

I've spent my entire career battling the type A vs type B personality dilemma since I am in the latter category. Not exactly social anxiety, but a near cousin in that it is very difficult to talk to folks and especially hard to sell yourself in an interview. Unfortunately many hiring managers seem more impressed with the type A individuals because they are the salesmen and tend to be more aggressive than type B which is actually a job requirement for them. As a result, the type B folks tend to get overlooked. It is not discrimination but rather the nature of some jobs. The key is to build up a story about yourself and your accomplishments that can be included in your resume. Let your resume and cover letter speak for you. Give facts with specific examples of your accomplishments, awards, etc which will demonstrate your talents. Focus on your ability to write, learn, research, use software, etc. Anything you see as your strengths. There are tons of jobs out there that a type B personality can do and utilize their strengths without having to face clients, customers or even a lot of other teammates. The trick is realizing that all type A people NEED the type B folks in order to succeed.
I am not sure about current practices, but when I applied for College (and later when my kids did), the screening was done based on your grades, SAT/ACT scores and an essay explaining your desires for the future. I would suggest using your time between now and graduation, looking for opportunities like Dana suggests in an earlier answer so that you have good material to work with when it comes time to write your essay. This is also the foundation and something you can build on when it is time to create your resume. Focus on what you CAN do and don't worry about your weaknesses. Most employers that are actually looking for type B people, are not concerned that you have the weakness, just that you recognize it and are either working on it or at least not letting it cripple you by focusing it too much. You are already demonstrating initiative by asking this question. Look for other examples where you have taken the initiative and document them. It is great material.
I actually started my College career with a totally different degree plan than I ended it with. I started working in a field which is totally different from what I am doing today. Always do your best, be willing to fail (since we all do), learn from your mistakes and you can be successful. I would love to be a Vet and actually got my degree in Animal Science. Now I work from home for a High tech company. Opportunity presents itself in all forms. Watch for your's. Good Luck!