In countries that have inherited the Westminster system, loss of supply typically results in government being required to either reform as the majority that blocked supply; or dissolve parliament so the electorate can pass judgement and presumably break the deadlock.
As far as I can tell, the United States doesn't have an exact equivalent.
So my question is: Does the United States have a built-in mechanism to resolve loss of supply? And if not, why not?
No the US has no mechanism for resolving this.
That's a consequence of two things that are distinct about the US system (compared with the Westminster and similar systems)
As DJClayworth correctly points out, there is no mechanism for forcing a resolution. But there are mechanisms for making a shutdown extremely unpleasant for Congress, ideally to the point that they pass a budget notwithstanding their political differences.
For starters, Article I Section 5 of the US Constitution provides that:
Each House shall be the judge of the elections, returns and qualifications of its own members, and a majority of each shall constitute a quorum to do business; but a smaller number may adjourn from day to day, and may be authorized to compel the attendance of absent members, in such manner, and under such penalties as each House may provide.
[...]
Neither House, during the session of Congress, shall, without the consent of the other, adjourn for more than three days, nor to any other place than that in which the two Houses shall be sitting.
Additionally, Article II Section 3 provides:
[The President] shall from time to time give to the Congress information of the state of the union, and recommend to their consideration such measures as he shall judge necessary and expedient; he may, on extraordinary occasions, convene both Houses, or either of them, and in case of disagreement between them, with respect to the time of adjournment, he may adjourn them to such time as he shall think proper; he shall receive ambassadors and other public ministers; he shall take care that the laws be faithfully executed, and shall commission all the officers of the United States.
The combined effect of these clauses is to prevent either House of Congress from adjourning without the consent of both the President and the other house. If a recalcitrant Congress refuses to pass a budget and purports to adjourn, the President can forcibly reconvene them.
Depending on the procedural rules of each house and the degree to which their respective members want to pass a budget, it is also possible to compel the attendance of individual Senators and Representatives, in the event their absence prevents passage of a bill. The Senate, for example, allows this by simple majority of "the Senators present". That has actually happened on multiple occassions.
With these mechanisms, individual concerned Congresspeople and/or the President can very well "lock Congress in a room" until they pass a budget. This does not, of course, guarantee a budget, but it does make one substantially more likely as time elapses, approval ratings fall, and Congress gets progressively more desperate to see the light of day again (or at least, to start campaigning about how the shutdown is the other party's fault). In this respect, it is vaguely similar to a Papal conclave.
The Federal government has not yet needed to take such drastic measures as these to pass a budget, in part because it is undignified for both the President and Congress. However, a similar sequence of events has recently happened at least once at the state level: In 2009, New York managed to completely bugger up control over its Senate, and the governor resorted to calling special sessions and even temporarily withholding the Senators' pay(!). It does not seem to have worked particularly well in this instance, though. The Senators for each party convened separate sessions, refused to recognize each other, and then immediately adjourned for lack of quorum.
