Selecting Project within a Form

Our team would like the ability, for the person completing a form, to select what Project the Task will be saved to. 

For example

We manage a lot of websites. We want to create ONE  "Website Edits Form" that our entire team can use. We want a field on the form so that the team member filling out the form can choose which Project the form will be saved to as a Task. 

Currently, as far as I understand it, we would need to either (1) Create a separate form for each website so that the form is saved to the website's corresponding Project. Or (2) We would need to have all the completed forms go to a single Project (like a holding pen), and the team member filling out the form would need to go to that Project and reassign the Task to the Website's corresponding Project. 

Both of these options take a lot of needless time.