"So how do we record those say travel expenses"

Start with How you Paid for it. s Expense from Checking (or on debit card) or as Credit Card charge.

"and keep track of to whom we must bill them?"

You split the details and assign Customer Name in the detail lines. Example:

Lodging split to 3 different Clients; as billable or not.

"For example, may I call the Vendor the Customer name"

No. That makes no sense. Example:

Postage from USPS is split as "postage reimbursed expense" by using your Service or noninventory product, listing that three times, so that you can allocate how much goes to each Customer name. USPS is your Payee name.

"and mark all costs as Reimbursable Expenses?  Is there a better way to record and track what I must bill each customer for?"

You enter something as Billable, to the customer, because you intend to Bill them for it. The same is true for Billable time, from timesheets.