Congress did enact a peacetime draft in 1940 before the American entry into WWII, with 12 months active duty until amended in 1941. The bill was first introduced in the senate as S. 4164 on June 20th, 1940, and referred to the Committee on Military Affairs, 4 days after Petain announced his intention to seek an armistice and two days before France signed the Armistice of Compiègne, so it was likely at least partially a reaction to the troubling speed of German advance, but I'm only speculating here.
While I doubt that any compulsory training and service will make it into law before American entry into WWI, it might only be a negotiating tactic so he can get a lesser expansion of military preparedness passed.
There is, however, a possibility for some form of national service to land right in Teddy's lap - courtesy of the Supreme Court and that pernicious "well-regulated militia" clause of the Second Amendment. For context, it wasn't until 2008, in DC vs Heller, that the SC ruled that previous limitations on gun ownership, even those predicated on the language of well-regulated militia did not prevent people from owning guns without any sort of militia membership. Presser v Illinois (1886) stated that the 2nd amendment does not restrict the laws of the states themselves, except in that they cannot deprive the federal government entirely of a citizenry armed so as to be available for calling to military service, and for a sense of how the militia clause was later used, US vs Miller (1939) asserted that weapons without clear purpose in a militia (like sawn-off shotguns) can be restricted in interstate commerce.
Hypothetically, with the right case brought before the Supreme Court, they might rule that the right to bear arms, for the provision of the common defense, requires that the bearers of arms be organized so as to be effective in the case of military necessity (and indeed, also so that they can't simply operate independently). Interestingly, Presser v Illinois offers a template: in that case, a socialist in Chicago was fined for leading a march of an unlicensed militia, and the charge was upheld by all the courts of appeal, with the SC ruling that Illinois, as a state, was permitted to put restrictions on the right to bear arms, while only the federal government was bound by the second amendment. What if a patriotic club marched with guns in DC? What if an armed socialist militia, again in DC, counter-marched? What if there's a clash, or arrests? Presser left open the question of gun rights in DC, and also left open a means by which gun rights in DC could be restricted, by using the second amendment itself. If socialist marchers are arrested while patriotic marchers are left alone, or if arrests happen well away from government buildings (eg, the socialists intercept the patriots several blocks before their march reaches the national mall, or perhaps even after the patriots leave the mall, to return to their headquarters) then charges of endangering the organs of government may well fall flat, and the case could reach the Supreme Court, where their only option to limiting socialist intimidation tactics would be to affirm the right of Congress to require the bearers of arms to be licensed members of regulated militias, since any other restriction (like requiring the bearers of arms to be "persons of good repute") would contravene the second amendment.
Teddy could use this to push for militia regulation that turns them into reserve forces with training etc.