The bottom of the form does not display on an iPhone or on an iPad.

http://a1apatio.com/contact/

It's not an iphone issue it's a webpage issue. You can see it do the same thing in FX on a pc if you minimize your browser window.

But why is it happening? What can I do to fix it?

Your responsive WP page has a break point at a width of 900px. When the view port is wider than 900, everything works fine, but when you go below 900px width, all your added content breaks the page. This includes the Google map, which completely disappears, and the fixed-sized form which moves up and on top of your contact text, which you have placed in a table.

What you need to do, to begin with, would be:
1. Make the form responsive for compatibility with the otherwise responsive page.
2. Read up on the details about how to add content to this particular, responsive, wp page.

If I had more time right now, I would have taken a closer look and suggested specific coding changes but, alas, I don't...

Per is correct and I see you have the form script inside a <p></p> tag. Better to place it inside a div and give the div an Id then apply the needed response css to make it lengthen with the screen reduction.

Although I am pretty comfortable with html, I'm not familiar with the coding changes necessary to implement your suggestions in WP. When you get some time Per, I would really appreciate some help with this.

I know it's not your job, but again, any assistance is greatly appreciated.

Thanks

This is only an example

Thanks, but where does that go?

You can place it in your style sheet or at the head of your page inside a style tag.