Is it possible to say that "someone does something as he or she can" to express that he or she is doing it with full enthusiasm or is this just badly translated German?
Edit: Unusual language wouldn't be a problem in this case.
The usual form is: "Doing X as best he or she can."
"As you can" is a legitimate phrase but it doesn't generally mean "with enthusiasm."
Please send me those reports as you can.
I do not know how pervasive this usage is, but its meaning is, "do this task when you are able" or "when you get some free time, do this task." You could reword it as, "when you can."
The more likely usage would be:
I eat healthy as I can.
This could be reworded as, "I do this whenever I can."
Other phrases that match the particular words "as I can" will be comparative:
I sleep as soundly as I can.
I exercise as often as I can
The last example again is meant to convey frequency, much like "I eat healthy as I can." I suspect that the latter example is a butchering of "as often as I can" and suggest explicitly using "as often."
I cannot think of a specific usage using "can" to suggest someone is doing a task with great enthusiasm. The other suggestions in the answers offer good alternatives with slightly different connotations. More typically, people will use a phrase such as what Hellion listed.
The short answer:
You could say that someone has:
set their mind to it, thrown themselves into it, or gone after it: given it their full focus and attention
put their back into it or given it their all: gone at it with their best efforts
done it as only he/she can: performed the task with a particular, identifiable style
gotten into it: displayed more and more interest/enthusiasm/obsession about the task as the work progressed
Another common form is doing X as only he or she can. This changes the meaning of the phrase depending on the context, but usually implies with enthusiasm.
You may also say "to do the best" - "he does his best" for an instance.