May I recommend
Freedback, as they do PHP forms that you can customize within their website. If you are FreeHosting, just signup with
789Mb. If you are paid, I'd seriously recommend transferring provider, but thats not always an option. Go to their site, design your form, and copy-paste the HTML. The PHPMailer is located on their server, so no there are no PHP files for you to host on your server, or the one through another company. I seriously recommend their services to all beginners.
Although the free service displays a submission advert, then goes away when submission is complete, its ideal for startups. I guess you cant be anything major, because you dont have a server with PHP support. I am bringing my new webhosting server online soon, so if you want in, you'd better PM me. 50GB space, unlimited bandwidth, 9Mb unmetered uplink and free technical support. PHP, HTML, MYSQL and so much more!
Anyway, try out
Freedback and see what you think. If you dont want to upgrade, but want to use PHP forms, their pricing options are quite reasonable, and they give you an option to redirect multiple forms, use password boxes, collect credit card details (within the Data Protection Act) and other stuff too.