Mogsdad got me on the right track, but in order for the submit button to work, I had to embed it directly in the code.
1) Create a form and in the forms menu, select "Embed form in a webpage"
2) Copy the entire block of code that appears. It should look similar to this:
3) Add this function to your script page
function launchForm() {
var embeddedHtml = ''
var htmlApp = HtmlService.createHtmlOutput(embeddedHtml)
SpreadsheetApp.getUi().showModalDialog(htmlApp, 'Title');
}
If you'd like the formatting and banner of your form to stay, remove the ?embedded=true off of the embedded link https://docs.google.com/forms/d/e/