How important is having research experience in the same department you are applying for? I have a B.S. in math from the United States and am returning to school in the US for a second B.S. in microbiology because: 1) I have no research experience; 2) I have no excellent LORs as a result of 1; 3) I can finish this degree in 3 semesters.

One would expect me to do research in microbiology / bioinformatics during this degree. However, I want to do graduate studies in statistics.

Is it frowned upon to go to the statistics department for research? On the other hand, would LORs from microbiology / bioinformatics have significant weight to statistics programs? When?

I'm asking because I need to be a student for legality purposes otherwise I would have to be paid, and I doubt that I would find a paid position. Microbiology is the fastest and cheapest way for me to get research experience.
If you want to apply to a doctoral program in statistics in the US and you have a BS in mathematics, just do it. There is no real need to take side trips into microbiology or anything else. Relatively few undergraduates in the US have any serious research experience when they apply to grad school. And in many graduate programs, serious research doesn't start immediately in any case. Some places have "warm ups" for the big event (dissertation research) but most have a lot of coursework leading up. 

And of course, until you apply, you don't really know your chances. 

Math is a pretty good background for statistics, of course. Much of the basis of stats is based on math. You may even have taken a probability and statistics course already. 

It is odd, only because of the strange path to a goal. The winding path isn't needed here. However, microbiology might be interesting if you actually have an independent interest in it. But it isn't likely to get you much closer to acceptance in a statistics grad program. 

Just. Do. It. 



For the record, in Europe this might be different. But that is because both the undergrad and graduate programs are quite different than in the US. 



And letters of recommendation is a separate issue, possibly serious. But it might be worth going back to your original math professors and talk to them about how they can help you. It is a hard thing to do for someone with autism, I know, but it is something you need to learn to be comfortable with to have a decent career. Hard doesn't mean impossible.