The official answer is no.  However, there is a solution you could pursue. To meet urgent needs, the Department of Defense (DOD) allows issuance of an undefinitized contract actions (UCA), which authorize contractors to begin work before reaching a final agreement on contract terms (DFARS 217.7401)  A word of caution, the contractor has little incentive to control costs during this period so you will want to definitize the UCA as soon as possible.  You will find specific guidance for UCAs/letter contracts provided at FAR 16.603 and DFARS 217.7403. 

Additionally, taking this route does requires specific documentation and approvals.  Details/specifics for pursuing a request for UCA approval package can be found at DFARS PGI 217.7404-1.  Good luck!