President Reagan in 1968 is a curious beast. We're a decade too early for supply-side economics so large tax cuts are out. Viet Nam is still ongoing, and with the conclusion of the Tet Offense both the Viet Cong and the North Vietnamese regulars are essentially crippled (in OTL, permanently for the Viet Cong). Domestically the Great Society ended as an expanding program with the 1966 midtems.
Let's go down the list. Reagan's Vietnamese options are not bad at the time. Rolling Thunder II (aka a reversion back to the pre-March '68 Rolling Thunder) is almost certainly a go, and with Congress (if not the Senate) quite conservative Reagan should be able to push forward with it.
With General Creighton Abrams's successful prosecution of the war IOTL it seems clear that strong political support and continuing high troop levels (535,000 in December, 1968) would lead to a reinvigorated South Vietnam—in the purely military sense—and additional North Vietnamese defeats.
As with any timeline where the US decides to win in Viet Nam the criteria for victory are limited. South Vietnam is a given and the Viet Cong should suffer the same fate as OTL but the North Vietnamese are unlikely to give up leading to a continuing low-level hot war and requiring an American military commitment on the order of Korea/Japan combined.
At what point can American troops be drawn down? I don't know. IOTL American airpower, funding, and weapons supplied to the South Vietnamese were enough for South Vietnamese regulars to beat their North Vietnamese counterparts. IOTL Watergate cut all three off, and South Vietnam lasted only a couple years.
A larger American force and continuing support should see South Vietnam survive for the foreseeable future. Could North Vietnam accept defeat? I'm unsure, but lean towards "not really".
As always the other main problem is South Vietnam's consistently horrible government.
Setting aside the military question, we return to political support. Can President Reagan win re-election with the war in Viet Nam going well? Where "well" is defined as winning, but with much higher troop levels than OTL 1972. This depends, naturally, on the Democratic contender and how well Reagan's domestic program goes.
As in Viet Nam, 1968 sees Reagan with a fairly wide range of options on the domestic front. Killing the Great Society, or some great part of it, is possible (probable?). Unlike Nixon, who essentially governed as a liberal in order to keep a free hand in foreign affairs, Reagan would probably take a somewhat more libertarian stance. However, as with his OTL Presidency (or his OTL Governor of California performance), this may not mean large reductions in domestic spending.
Reagan in '68 may raise taxes to aim for a balanced budget, and as Nixon attempted may enact some sort of negative income tax in order to eliminate as many entitlement programs[1] as possible. Trading—to Congress—a guaranteed annual income (a liberal wet dream) in order to kill Social Security, Welfare, and so forth would be both bold and incredibly tempting if he was willing to stake his Presidency on it. '68 Reagan may be more willing to take bold gambles being both younger and closer to the Goldwater libertarians.
Finally we turn to social issues. Roe Vs. Wade is looming but for the first term it's all about civil rights. Whereas Nixon exploited the issue for political gain it's quite possible that Reagan takes a stand against it on libertarian grounds (school busing, in particular) or enforces it to the limited extent possible. This won't hurt him in the South—obviously—and would probably prove popular with a number of people elsewhere.
A side effect of ending school busing would be American cities that remained more viable than IOTL. How much would this radicalize the urban black population? Would radical black action swing the Silent Majority firmly into Reagan's camp?
Alternatively would Reagan take another gamble and support civil rights? This would hurt in the South (Texas, for instance, will probably remain Democratic a little longer) but not fatally given yet-stronger Democratic support and may convince some segment of the black population in the Northern cities to consider voting Republican again.
1968 was a key year. Viet Nam, civil rights, blue collars workers/unions, the future history of both political parties, and pretty much the whole culture war (not to mention most political leaders) was born in and around there or that year featured as a major turning point.
President Reagan, in other words, has a vast number of options open to him.
[1] Interestingly Nixon created most entitlement programs by indexing them to inflation, Reagan would almost certainly not take such a step regardless of what else happens. This would leave future US budgeting in vastly better health.