"very willingly" is correct and can be used in written English.
It is an adverb phrase that means "with a great amount of willingness or enthusiasm." Example: She agreed to help her friend move out very willingly, even though it was on short notice.
And we do so very willingly.
The answers have not come quickly or completely — and not very willingly.
That is something he will do very willingly, White said yesterday.
I was in the wrong place at the right time and very willingly offered my services, hoping for the very best for the future of our hometown.
He added: "If you had asked what has surprised me in the last few weeks, it would have been that this has been accepted very willingly.
About their son's conception and birth, she wrote in a memoir, "Having not participated very willingly in this future event, I was perturbed".
"Very few athletes willingly retire from professional football.
The bus scenes are one example of how the audience at a good play – and Circles is very good – will willingly supply the pictures that the scenery doesn't.
Then they will come and they will willingly work very hard, over and above what is expected or paid for.
Flanked by advertisements for an Arabian stallion and a cobbler's apprentice, a short item read: "A Gentleman about 30 Years of Age, that says he had a Very Good Estate, would willingly Match himself to some Good Young Gentlewoman that has a Fortune of £3,000 .No record survives of the response, if any, to what is thought to be the first lonely-hearts advertisement.
I made a new friend who invited me to have dinner in her home, people who took the time out of their busy schedules to show me their cities, and just very kind people who willingly, on their own, lent a helping hand.
