Anyone who tells you you can’t make strength gains while being a vegetarian is either
a) an idiot
b) spouting off bullshit they heard on a web forum
c) trying to sell you something. Probably meat.
Those are solid goals, but you can and should be able to make much better gains in the squat by the end of the summer (12 weeks). Probably closer to squatting 205.
What you’ll need is a solid training program. I recommend Rippetoes, because it’s created those kind of results for others on this forum. If you’ve been doing Rippetoes already, up the volume to Madcows 5x5 or Rippetoe’s Texas method (also a 5x5).
Being vegetarian is no excuse to not be getting a ton of protein. Whey is cheap and effective. If you’re vegan, pea and soy protein will also work. Nuts, beans, and pasta are all also decent in the protein department. make sure you’re getting a gram of protein each day per pound of bodyweight (150 right now, more as the summer goes on and you get bigger).
Also, make sure you’re packing in the calories needed to fuel growth. Bodyweight x 20 in calories is generally recommended. Which means 3000kcal each day. Use a food log to make sure you’re getting enough. If you’re not sure, err on the side of eating too much. The cleaner your food choices are, the better your gains will be, but a little junk can go a long way.
Vroom’s thread at the top of the beginner’s forum has an index of good nutrition articles by John Berardi. Read them if you haven’t.
Good luck, keep us posted with your progress