I’m currently a high school sophomore taking AP Calc BC, and next year I’ll be taking Multivariable Calc, but I don’t know what I’m going to take after that. The only math course available to me after that is AP Stat, and I don’t want to take that in place of a calculus course. What do I do?
Does your school have Differential Equations (or is that included in your Multivariable Calc class)? You also might want to consider dual credit taking college math if you have the option, transportation, possible funds, etc. At my school, Differential Equations and Multivariable Calculus are the highest courses offered in the school, and they’re each a semester long. Btw I’m really impressed with the difficulty of math you’re already at as a sophomore. 
So, you’re planning for senior year. Will you be able to drive to a community college by then?
You’d probably take Discrete Math and Linear Algebra after that, BTW.
Ordinary Diff Eq, Linear/Matirx algebra, and calc based stats are all solid dual enrollment options. If you don’t want to do that I would take AP Stats (math), AP Physics C (calc based science), AP Computet Science A (technically math according to college board though this is iffy) and/or be a TA (teacher’s assistant) for a math teacher in your senior year so you don’t loose math skills. Dual enrollment is the best option though. What do kids in similar postions at your school do?
Just take AP Stats! You won’t regret it. 
In my school there’s a thing called complex analysis, but it’s not AP class.
Can you take classes at a local college? If so, what would you be able to take?
At this level, there isn’t any particular progression you have to follow, and you can even take multiple math classes at once if you have room in your schedule. At my university, someone who has taken multivariable calculus is eligible for quite a few undergraduate math classes (real analysis, dynamical systems, knot theory, probability, etc.). The most common things to take are probably linear algebra, differential equations, discrete math, and/or calculus-based statistics. You should consider your interests and maybe the requirements for your intended major.