Well I suggest you look at other services that already do this. I can tell you that some of your competitors already do this with simple IFrames or using simple JS scripts that a person can simply copy and paste on their page. I don’t think the back end tech really matters all that much, so whatever is comfortable for you would be fine (I guess PHP in your case). But I know of other services that use Ruby (like Ruby on Rails) or Python (using Django).
But again, the main focus should be integration options for your customers and there you can do the two technologies I mentioned. They are easy to implement, supported in most browsers (if not all), are usually enabled automatically and provides a ton of flexibility in managing forms from your side. For instance if they are iframing, then it is your form they are inlining to their site. In the case of JS, it is your script they are usually referring to so you can code that however you want.
I hope that helps.