When applying for a job, how would I make it clear that I can't do phone interviews due to speech problems?

Bear in mind, U.S. Federal Law prohibits them from ruling you out as a candidate solely on this disability, via the Americans with Disabilities Act, as long as you are still capable of performing essential duties. (So, you couldn't take a job in phone tech support, because talking on the phone is an essential duty, and that's legal.) Talking on the phone is seldom, if ever, an essential duty for a programming job.

In the initial written communication, tell them what you told us: because of a speech disability (be sure to use that word, it flags HR to legal trouble if they chuck your resume on those grounds), you cannot do voice-only communication, but face-to-face communication IS still viable with reasonable accommodations.

If they turn that into a deal-breaker, that's a red flag that they are (a) not sensitive to disabilities, and (b) not afraid of running afoul of Federal law, and therefore you wouldn't want to work for them anyway.

I'd like to start off by saying, because it's not said nearly enough: your handicap does not make you less worthy or talented than any other candidate, and you deserve to be in our industry.

In terms of the day-to-day, unless a team is mostly remote or communicates largely via phone, then as you mentioned, I don't see how this interferes, especially if you're still able to easily communicate face to face. Thinking in the context of my current team: We have remote Fridays where we usually do a stand-up via phone. If phone conversation was difficult for someone, I'd just ask them to send their update via e-mail or IM during the session so someone could either read it or so that appropriate follow-up conversations could happen as needed.

In terms of getting hired, my guess is that you've had to (and will continue to have to) deal with bias, whether it's unconscious or not. I'd suggest going one of two routes: 1) make the speech difficulty clear up front or 2) wait until you've already scheduled an interview to disclose.

Stating it up front could clearly communicate which is good, but could run the risk of a company passing you over for that reason (ugh). If that happens though, chances are they're a terrible team anyway. Waiting until after you've secured an interview (via e-mail etc.) may help in that someone has already selected you prior to knowing anything about your difficulty, which would hopefully lessen any up-front bias.

If phone is difficult for you but videochat is less difficult, you might also suggest having the initial meetings via Google Hangouts / Zoom / JoinMe or some other video service. The additional advantage here is that it demonstrates you understand communication is more than voice anyway (which it is).

I know this is just one random dude's take, but I wish you the best of luck with your job search!