I recommend using a web search on the following:

irs stimulus faq

Because the IRS has the answers and are doing a good job of keeping their FAQ updated as the details get defined.

For example, right now I see question 58:

https://www.irs.gov/coronavirus/economic-impact-payment-information-center#ssrrva

I looked all over IRS website. Still couldn’t find anything referencing my situation. 

"I looked all over IRS website."

I gave the link, and I pointed out it is question 58. You don't need to look everywhere; just read the answers they provide at the question that matches your question.

I don't see any reference to married filing separately there?

"I don't see any reference to married filing separately there?"

Question 58 is for Social Security, Railroad Retirement and Department of Veteran Affairs benefit recipients. That's what you asked about.

First, it tells you there is Nothing needed to be done for the client, the Beneficiary, as you describe. The client that gets SS direct deposit would get the stimulus DD as well; otherwise, as a paper check.

Second, it explains how filers will get their payment, which is the scenario for the spouse you describe.

They are both explained there.