I wasn't really going to place another answer here, but I feel that some things must be said, as I've read some misunderstandings that my conscience can really not leave be. You need to work on the maturation of certain interpersonal and behavioral patterns.
My colleagues also feel I am being bullied by the one specific individual who is causing all of this. They feel the trigger is that...
Communication has never been served well just by "feelings", they are subjective and, usually, based on a lot of past experience that others don't necessarily share with us. Second-guessing is a great way to lose time, ruminate unproductively and feed your bad feelings. I know it's hard, but if you would like to know why, just ask. There have been times I have been getting on someone's nerves without even knowing. God, I wish they had just let me know earlier, I felt such an idiot after finding out how my actions were causing them distress. Be prepared, however, to not always be able to resolve things by asking... immature behavior is something you will have to be able to deal with, and, occasionally, the answers may not be meaningful, or even make sense. That is especially common when immature competitiveness prevails and stakes are at place.
I cannot handle the anxiety because...
You don't need to feel like you are "explaining yourself". You should get used to being considered innocent until proven otherwise. You are not expected to be able to handle any unreasonable level of anxiety all by yourself. Even in anxiety-riddled jobs, like law enforcement, your training has a dedicated focus on that fact of life. Furthermore, anxiety is not (easily) objectively countable, what may be causing you an X amount of anxiety, may be causing someone else 10 times more anxiety. We are different, and there are measures in place to ease the coordination of those differences.
If you are feeling a sudden increase in anxiety, stop there, take a moment and check it out. You are not alone in anything in life, you can ask for support in handling this, just like you can ask for support in handling anything else. If you don't know who to go to for support, you can always ask that. Make it a habit to build your supporting infrastructure and "defenses" before everything else when starting any job (or any other official relation for that matter). This includes you knowing your rights, how to resolve or handle disputes, who to go when you cannot handle something, etc. It may come to some people as a surprise, but questions like "who to go to when in trouble" are not stupid or funny, especially regarding professional environments; they are very serious and meaningful, if you are aiming for a healthy and productive workplace culture, at least.
I am afraid to share this with my boss because I do not want to appear weak...
Newsflash for you! Sharing reflects strength, not weakness. You need to be strong to share, so you got this wrong. Any boss regarding this as a weakness needs to step down or get their story straight. Weakness is a qualitative feature that is implicated in reduced action. Speaking and sharing is action, not lack thereof.
[...] or unable to handle criticism/the competitive nature of Software Engineering.
How well can you handle criticism? A good result in any professional effort has criticism as a prerequisite, how else are you going to find about your mistakes and improve? Fair and efficient criticism stands firmly on de-personalisation of the conveyed content, it is not driven by emotion and should not be interpreted as such. If you are not able to handle criticism, then either you are misunderstanding what criticism is, and possibly need to work a bit on your attitude, or, as is most likely in your case, what you are receiving is actually not criticism but aggression.
Another colleague was bullied by this group, got an accommodation to work 75% at home due to his anxiety from the situation before a spot opened up on a different team.
So the company assists in marginalising employees, based on whatever rationale some group plays upon. If you feel insulted by this mentality, good, you are totally right. Do you want to keep working there? If yes, then you need to gather some of those people that are on your side, and speak up, follow the proper course, exactly as Anthony exceptionally put it. Keep in mind that this effort is also going to benefit the company (if they care about a respectable workplace culture, that is), so you should be proud of taking this initiative, which, once again, demonstrates strength, not weakness. If not, I can assure you that competent web developers are on high demand, worldwide. You can demand high standards from your employers, just as they do so of you.
Would it look bad for me if I sought out the same accommodation just to get through how ever long I am on this team for?
Imagine a parent, listening to their child, in the early morning, saying that they "don't want to go to school today", because some guy is picking on them for no reason, and being there is like hell. You are effectively asking whether it would look bad on the child to go to school only half of the days (effectively missing out on communication, collaboration, shared knowledge, fun, etc), until school is over. Who would even contemplate for a moment that this could be a solution? I know the comparison feels harsh but it is what it is. Bullying is bullying for adults as much as it is for minors. To directly answer your question, it would not look bad on you, not at all. You know why? Because those forcing you to ask such accommodation cannot really feel anything, let alone good or bad.
I am concerned that it’ll make me look like I am in cahoots with my colleague who was bullied by this team.
But you should be in cahoots... on how to break this workplace habit of this team. Ask your colleague what they went through, help them follow the course set forth by Anthony's great answer. Joint effort against bullying in the workplace is frankly nothing to be ashamed of.
HR is not your friend
I've seen this thrown around and I must say I strongly disagree. Perhaps treat HR as if they were not your friend, but do not count on them not being able to be your friend. Managing to align a significant part of your company's interests to those of your employees is a truly wonderful, however elusive, workplace experience.