Most of the shops I have seen say that you cannot use a coupon or Groupon or Restaurant.com thing, but most of them say you can use a gift certificate. It is always safest to read the guidelines for your specific shop and if you have a wonderful coupon and the guidelines are silent, contact your scheduler to ask.
But here is the deal. If I am being reimbursed $35, I really don't want to buy and report on $45 of food, use my $10 coupon or gift card and only get reimbursed $35. Similarly if I buy $35 worth of food I don't want to 'waste' my $10 coupon and get reimbursed only $25.
So, if I had a $200 check at High End Steak House, I could pay with the discounted gift cards, saving about $28, and still receive the full reimbursement from the company, if payment with gift cards is allowed.