Hi Victoria - have you thought about using SurveyMonkey? On the web, they are at www.surveymonkey.com.
I'm just kidding... I did read your post. SMS-based surveys are more complicated than you'd expect (or rather, they get very complicated as soon as you want them to be interactive; the web is much closer to a piece of paper, which is what people usually think of as a survey) and the market is kind of small, so there have not been a lot of tools developed for it. We built our own at one point, but abandoned it with the rise of smartphones.
The biggest market for it currently is non-profits in developing world, but even there smartphones are slowly taking over (well - smartphones have taken over, but data is still not widely used, so SMS still rules there). Some tools have been built on an open source project called RapidSMS (https://www.rapidsms.org/projects/). I doubt these will work for you exactly and I doubt you want to get into the business of maintaining your own code for SMS surveys. However, try networking to some of the people you see connected to that project through LinkedIn, and they will likely know of a reasonably priced paid solution for you. They do this sort of thing everyday, just not in the US - so the technology is out there, and has been out there for a long time at this point; it's just not popular.
If you can reduce your "survey" to just one or two multiple choice questions, you can potentially use a poll instead of a survey: http://www.smspoll.net/ However, it's not the intended usage... so it's likely not going to work exactly like you want.
Alternatively, you could re-purpose a call-flow system to conduct a survey:
Even though what you want is pretty basic, the problem you are going to face is that few people really need it at this point... So you're going to have to be looking for a small business that does this sort of thing (meaning you need to find them by networking) or find a more complicated solution aimed at a broader problem you can simplify.
There are larger companies that do market research like Qualtrics and Mosio that offer niche technology solutions... but their pricing may be prohibitive for your application. You might try to find someone like this: https://www.txtimpact.com/text-message-survey/ who is clearly a much smaller player. Never heard of them though - no idea if they are reliable.