Without using light vehicles?
That's a fair remark, Elchino responded already yes today Ostheer can push from minute 0 with assgrenadier and the amount of players using them shows that they are effective and Ostheer doesn't suffer from not having snares early game (at least vs USF).
Then Grenadier are weaker than riflmen for the first 3-4 minutes, once you upgrade the lmg42 they beat vanilla riflemen. If you want the riflemen to stay on top you must paid 15 fuel. If you want the riflemen to also use grenade (as grenadiers do) you must again pay 15 fuel. If you want to heal them, pay 10 fuel more.
At the same time you get your first lmg42 you also get your first pzgrenadier which is completely capable of pushing anything 1:1 the USF player can oppose, after all that's an elite unit.
Now we can't take the Gren vs riflemen matchup in a vacuum, there is also the HMG42 and pioneer with super vision to articulate your early game.
So yes, I don't see any particular reason for Ostheer to not push only with infantry.