I am a US Government contracting Officer. You are right, the term "let a contract" or "let" has no legal definition and therefore no legal standing. The term we are required to use is "award" or "award of a contract". Contracting professionals rarely use the incorrect term, however, customers or the requiring activity nearly always use the incorrect term "let". This may seem rhetorical but "award" has a legal definition. So we as contracting professionals must educate our customers and advise them of the correct terminology.
I worked in contract administration in a State Highway Agency for many years and the term "let" was used frequently. I checked a few legal dictionaries and one said "allow" which is just the common def. Of let. The other said "to award a contract". To me, awarding a contract means the work was advertised, bids were submitted by contractors, bid were analyzed, bidders were determined to be legitimate (responsible), the successful bidder signed all the nec. Documents (the contract), and the State said to the contractor--This is your contract and you are to begin work on______ (a date about two weeks in the future). Maybe let comes from: Let the contract work begin. The last part is strictly a guess by me. Others speaking in general might say--The State let a contract to build a bridge right over there. But that could mean many diff. Things. It could mean anything from design, advertised, accepted bids, opened bids, verified bids, awarded, signed documents or gave notice to proceed. My state had terms for every step of a contract. Let was not one of the official terms.
Define of Agreement