Just so I understand, your firm is just fronting the cash for the vendor's services, correct? Your client is going to pay the full amount of the cost later when you invoice them. Do I have that correctly? The trick is not to book to an expense or revenue account so you don't over/understate your net income.
You can do it with an asset account called Unbilled Client Costs or something similar.
When you pay the vendor for the services (let's assume you pay the vendor in cash):
DR Unbilled Client Costs
CR Cash
When you bill the client for the cost:
DR Accounts Receivable
CR Unbilled Client Costs
When the client reimburses you for the cost (let's assume the client pays you in cash):
DR Cash
CR Accounts Receivable