How much experience do you have as an ASM? Where?
You can send your resume to the PSM at the shows you wish to work on, however, because they are union positions, I wouldn't hold my breathe. You can make great contacts and if you shadow that would be the best way to make an entry.
Try to get on a show as the PA. See what shows are about to go into rehearsal and contact the General Management office to see if they have the PAs in place. If you are good you will establish a relationship with that office and that PSM.
It's best to get some experience under your belt before going for Broadway. Colleges tell you go for it but it can be a tr ap if ou aren't sure what you are doing. Get some non-union touring on your resume and then go for it.
Good luck