I'm the happiest I've been in months rn because my crush just broke up with her boyfriend
Why would this make you happy?
Because I can ask her out and she'll accept my invite instead of his.
I get that you are happy about it, but you should not move too quick. If a break up is fresh, she may not be ready to move on.
I'm not interested in having a relationship with her, I just want sex.
Even if you don't want anything serious, your crush may be in a vulnerable emotional state right now. It would be crappy of you to take advantage of that and try to have sex with her now. You should wait until the dust settles and she's ready to move on.