Well, it's more about your risk tolerance than anything else.
I'm not sure if IRS knows your immigration status and even if they do, do they report "suspicious" earnings to ICE and if they do, does ICE goes out and investigates such cases?
So that is your unknown and you can simply risk it: register a business as DBA, open a Stripe account or PayPal account and start charging for you saas and report that as an income to IRS just like you would normally do.
The worst that can happen is that you'll get deported. I googled "deported for having a company on h1b" and nothing came up so there's that.
If you want to lessen the risk, you can register the company outside of US. Either in your home country (additionally, you could have e.g. family member be an owner, to completely cut connection to you) or in some jurisdictions that allow opening companies without being there physically.
That's still probably not entirely kosher but harder to trace.
To be completely safe, you can launch your project as completely free, build it up, wait until you get green card and then start charging for it.
There was some discussion in https://news.ycombinator.com/item?id=3691541
