Figured it out by debugging, line-by-line, the SheepIt code!
This line was what was messing me up:
<select name='etypes[#index#]' id='etypes-form_#index#_etype' class='form-control auto-width inline'>
SheepIt was looking for it by ID, but using the name of the form instead: emails
Updating that line to the following made everything work wonderfully:
<select name='etypes[#index#]' id='emails-form_#index#_etype' class='form-control auto-width inline'>