Yes. You can submit as many SBIR and STTR proposals you want to different agencies. In fact, many of our startups use the core content of one proposal to apply for a number of different solicitations. There is even strong data that shows that most of the successful awardees have utilized multiple submission as a strategy to win.
However, you should note that you CANNOT double-dip for the exact same work from more than one agency. So if you do win two SBIRs from different agencies for the exact same work, then you will be asked to either pick one or modify your scope of work in a way that shows clear separation between the awards.
During proposal preparation, your proposal will naturally be tweaked differently for different agencies, because all the agencies care about their own unique missions and so will unlikely fund the exact same work. For example, the NIH cares about the health outcomes of a potential new drug treatment, but the NSF will care more about the platform technology used to develop the treatment, esepcially if it is highly innovative and applicable to a number of diseases. Similarly, DoD (and its components) specifically fund dual-use technologies that have applications within defense.
The TurboSBIR team can help you with this positioning for specific agencies. Read more about our proposal writing and consulting options here.