Hello!
I was thinking about your needs within your budget range and I have few thoughts and offers on it. In my opinion, the best form/site like that could be achieved with Bootstrap framework. Its fast, responsive, and already includes a lot of beautifully and cleanly designed form elements that can be added to the site really fast. It also includes jQuery, a javascript library with huge amounts of free plugins that could add powerful and modern functionality to your form. It would also be much cheaper to upgrade if you demands to this project increase because developers who know Bootstrap and jQuery are very common. If in future you decide to add unique design to your site then there are a lot of cheap bootstrap templates out there that are very easy to install.
Since i am new in this site i am eager to work with a low budget to get reviews and increase my reputation. Besides that, I have 10 years of experience creating and upgrading websites. I dont have info on how complex the whle site/form would be but I guess, with Bootstrap, I could get it up and running with few days.
I hope all works out for you.
Best regards,
TaunoM