Hi Tiffany,
Since your organization has Forms, they should also have a license for the "Avante Forms Portal", with which you can have your employees access the form without each having a named user license.
As far as I know, the easiest way to format/populate the reimbursement form according to the travel request form would be to create it as part of the same process using the same fields/variables as the travel request form (this will cause the reimbursement form to contain the same values as those that were originally input on the travel request form).
Then, you could setup your process to include an 'Email Service Task' that would trigger according to, say, the projected return date (which I imagine would be a field on the travel request form). When the return date passes, an email with a link to the reimbursement form would be sent to the employee that initially submitted the travel request. They would fill it out and submit it; it would then be available for your review, filing in the repository, passing to Workflow, etc.
I hope this helps, let me know if I can clarify anything.
Rob