The structure of this question is kind of awkward. For example, let's say these were all the options. Are you expecting an answer that's just, "Yes, those are all the options"? What I think your actual question is is "what is a good software architecture for your problem".

That said, you haven't really provided enough information to answer that question, at least not clearly. Do the differences between programs actually lead to differences in the structure of the tables? Does a table for program A have different columns from an analogous table for program B? Is it possible to make a reasonable design where creating a new program is as simple as inserting some rows? I would assume not, but the requirements you listed don't jump out at me as things that should require different database structures.

The structure of this question is kind of awkward. [..] What I think your actual question is is "what is a good software architecture for your problem".

You are right, and this is not a good question, but one that reflects my ignorance in this topic. Also, your suggestion is right, but spending years on Stackexchange sites, this variant didn't even occurred to me, because questions of this form get closed or deleted right away. (Well, I posted this there too, but got closed and deleted within hours anyway, so thank you for taking the time for providing constructive feedback:)

you haven't really provided enough information to answer that question, at least not clearly.

I didn't want to inundate readers with the details when the database structures are indeed fundamentally different (should have spelled this out probably), and the only entities common are the clients and employees.

Do the differences between programs actually lead to differences in the structure of the tables?

Yes. One may be able to coerce one or two table to serve multiple purpose, but it would be a mess. (Talking from experience as this is what the 2 previous "professional" devs did, after we decided to do this in-house...)

Does a table for program A have different columns from an analogous table for program B? Is it possible to make a reasonable design where creating a new program is as simple as inserting some rows? I would assume not, but the requirements you listed don't jump out at me as things that should require different database structures.

The programs are different enough that these gymnastics will lead to a mess.

Again, I appreciate your questions because it is now clear that if I can reformulate this question by providing the right amount of extra details to make it clear, I'll probably be able to answer my own question..:)