With all the misogyny hoopla surrounding the Elliot Rogers story, I'm surprised someone hasn't scolded you for using the phrase "get a girl." (Don’t worry about it BTW, it doesn’t mean anything)
The way to get a girl to go with you to the prom (at this late date) is to simply ask one or more until one says yes...recognizing, of course, that none may do so, not necessarily because they wouldn't want to but it is late in the game.
Getting past the rejection associated with a girl turning you down is easier said than done, but as others have advised you, it really isn't the end of the world, and once someone says yes (and eventually they will) you'll pretty much forget it.
You don't need a "plan" to get a date for this prom beyond asking as many girls as it takes. I think it's stupid that you should be pressured into going but since you've declared you don't have a choice, the blitzkrieg approach is your only option. If you strike out completely don't despair, and I agree with whoever suggested that you go to the prom solo. I wouldn't be surprised if there are girls going solo too and, if so, you will have a pretty ready-made situation for talking to one or more - providing you push yourself and don't hang out on the sidelines all night.
Now, as for a "plan" to get a girlfriend. Assuming you want one there are things you can do to improve your chances.
Don't sell yourself short. Make an honest list of all your good qualities. They should be a source of confidence for you.
Work on improving yourself (you should be doing this anyway). Make an honest list of those areas which might be an impediment and seriously try and do something about them. Some boys your age are not particularly concerned by how well groomed or dressed they may be. If you're not washing or combing your hair every day, this can easily turn girls off. We have no idea what you look like but there are simple things you can do to improve your appearance. Keeping clean and well groomed for example is pretty easy. Wearing clothes that fit you isn't particularly hard either. You don't have to be wearing whatever the latest fashion for kids your age may now be, but you probably shouldn't be wearing clothes that were in style 10 years ago. A button down shirt or golf shirt is always in style and unless you are attracted to girls who are Goths should serve you well enough. Chinos or jeans are fine too, if they are clean. The key is to look like you care about you look.
When I was in school it was amazing to me how many girls and boys were considered really good looking who, in my opinion, really were not. What they had going for themselves though was confidence. They thought they looked good and so a lot of other kids did too. They also dressed and groomed themselves as if they cared about how they looked. I suppose there are girls who are attracted to guys who dress like slobs and have greasy, stringy hair, but I never met any of them and I was a hippie in HS. The girls who found my long hair attractive wouldn't have done so if I never washed it.
The basic stuff like brushing your teeth and wearing deodorant seem like no-brainers but a lot of guys your age don't keep up with them
Obviously it's not all about looks, but your appearance can and should speak to your confidence.
Be a nice guy, to everyone. Most girls don't find bullies or smart-assess attractive, and no matter how nice you are to them, if they catch you being mean to someone else, you're probably dropping out of consideration.
Among your list of strengths may be qualities like funny, or smart, or creative. These qualities can be very attractive to girls. Most girls like guys who can make them laugh so if you're funny, that's a plus. Being funny is not being a clown though nor is it being really good at sarcastic cut downs. Being smart about a lot of things including practical matters can be very attractive, and being creative can be a superpower if you are capable of using it to show a girl you think they are special. But any of your strengths will be attractive to girls if they are employed in a positive way and boost your confidence.
Next, force yourself to talk to girls. Not about going out on a date but about anything that can make for an interesting conversation. This can be difficult at first, but it will get easier with time, and especially if you are not focused on asking them out. Things happen in school all the time that can serve as the ice-breaker, and then just go with the flow from there. Not every girl is going to enjoy talking to you, nor are you going to enjoy talking to every girl, but talking to a lot of girls will help you determine which ones most interest you, beyond simply their physical appearance. And you will be surprised how much better someone can look when they laugh at your jokes or share your interests. This last bit is important. You're likely to find a girlfriend in a situation in which your interests are at play. You say you're good at science, well I'm sure there are girls in your school who are good at it too. Unless school has changed a whole lot from when I was in it, there must be some extra-curricular activities that are based on science: clubs, a science fair, etc. A girl who is involved in these has already indicated that she shares one of your interests and talking about something you like and are good at is always a lot easier than talking about something in which you have no interest or aptitude.
You don't have to have a girlfriend. It's great to have one and they are a lot of fun, but if you don't get one right away, don't sweat it. Just keep laying the groundwork, by being yourself when your self is attractive, by improving yourself when yourself may not be, and communicating with people - especially with girls. After doing this there's a real good chance that you will have no problem identifying the girls you want to date and who will want to date you. If you ask a girl out with whom you have been sharing enjoyable conversations and laughs for a while, the chances are pretty good that she will be waiting for the question and say yes. But if she doesn't, roll with it. You can still have those conversations and laughs. Being friends with a girl is worth it even if she doesn't become your girlfriend.
This next bit could be controversial here, but if the girl you want to date says no, it doesn't necessarily mean never. I'm sure as hell not saying you should harass her or God forbid stalk her, but there's nothing wrong with waiting a while and then asking her again. If she says no twice, that's probably it for her, but it really can't hurt to ask a girl out twice.
Good luck for the prom (you're going to need it I'm afraid) but let us know how it works out and good luck for your future as well. You're in a great period of your life and you should be looking forward to having a lot of fun with and without girls.