Become.

If you need creds, I have a doctorate in linguistics and I used to work for the dictionary ...

I would write it like this:

They have expressed a desire to apply once programs become available in their communities.

If it ALL had happened in the past (i.e. they had expressed the desire AND the programs had already become available), then I would go with became. But in your case, I think it should be become. It is not past subjunctive but rather future/uncertain subjective, right? (Kind of like "if I were a rich man"?)

Become.

Became would indicate that they are available at the time you are writing this.

Generally, you want to stick to the same tense within a sentence. So, "became." Or change "expressed" to "express" and use "become."

*sentence*

another vote for "become"
think of it this way: your sentance could also be:

"They expressed a desire to apply when programs will (BECAME/BECOME) available in their communities."

it means the same thing and you wouldn't say "will became" available.

Become.

Another example:

Yesterday or today, that movie you hate became available on DVD.

Tomorrow, your favorite movie will BECOME available on DVD.

BECOME!!!!!!! - became is past tense - they aren't planning to apply when the programs "became" available - see. changing the order of the words makes it more obvious.

If you were talking in the past tense, that someone previously applied for something when the programs "became" available - again, this works when talking about the past - not the present or future.

Wanted to add that it would be "become" if expressed changed to "express".

Become, if the programs are not yet available.

Agree with pp. It's "became".

Became available.

Past subjunctive tense.

I've gone back and forth on this sentence, and I can't figure out which verb tense for "become" is appropriate:

"They expressed a desire to apply when programs (BECAME/BECOME) available in their communities."

The programs are not yet available, if that helps figuring the right tense. Any insight? Thanks.