"onward with" is correct and usable in written English.
You can use "onward with" when you are expressing that something should continue. For example, "Onward with our mission to help those in need!".
Onward with the revolution".
We move onward with the years.
Europe can now urge him onward with improved trade relations.
They stayed three days, then marched onward with the porters.
Now the N.H.L lunges onward with a full half of its season remaining.
The depression builds, but I will continue onward with my research.
But you should continue onward with a cheerful wave; it's just a part of suburban driving etiquette.
The moment passed quickly, and then he was back in gear on the Met stage, stomping onward with composure.
But again and again it was she who pushed ahead, at one point flicking her partner's feet onward with hers.
These states capitalized on trade both with peoples of the savanna and, from the late 15th century onward, with Europeans.
I didn't manage my life from the age of 18 onward with the intention of being a Family Values US senator.
