The only time you can force a competitor to do anything is if they are interfering with an existing contractual relationship that you already possess. In your scenario, this is not the case. The real question this scenario seems to be asking is "how can the government be required to enforce its own regulations" or more likely, "it's own RFP". You posit that the widget materials requirement appears to be statutory or regulatory under your scenario... At least as I read it.
This would more likely be spelled out in the RFP. A request for proposal (RFP) is a solicitation, through a bidding process, by a government agency, municipal, state, or federal, interested in procurement of a commodity, service or valuable asset, to potential suppliers to submit business proposal. This is typically where you find any materials requirements.
Government procurement law sets forth several requirements to combat against bid rigging with specific oversight mechanisms depending on whether we are discussing local, state or federal contracts. Generally, government agencies do not ignore the requirements that they themselves set forth. These are usually in their bidding procedures and guidelines, such that when they put a job or a contract out for bid ( the RFP), guidelines exist to ensure that the contract award goes to the most qualified, lowest bidder who can meet all requirements of the procurement request. There is oversight built into these strict public procurement laws.
However you seem to suggest, in your hypo, that the government agency is ignoring their own proposal for material requisites, or the regulation that informs those requirements. If this ever occurred, the best you could do is complain first to the head of the agency who put the contract out for bid. If they ignored you and you had legitimate proof their widgets were substandard, you would complain to your government representative for your district, to seek enforcement under whatever statutory or regulatory framework those material requirements would fall under, if not the RFP itself.