'flurry of emails' is correct and can be used in written English. You can use it to describe a situation when a lot of emails were sent or received over a short period of time. For example, "I received a flurry of emails yesterday with updates on my project."
I woke to a flurry of emails about the news.
After an initial flurry of emails we spoke on the phone.
England's squad should anticipate a flurry of emails containing attachments featuring instructive video clips.
After a flurry of emails, Iceland pledges to guarantee bank deposits up to maximum of €20,887.
He sent a flurry of emails to staff at the film's makers, Sony Pictures and Columbia Pictures, and received a reply from a former set designer on the film – two years later.
And after the flurry of emails and phone calls ends, ask your counterpart to confirm that you have indeed reached a deal that satisfies you both and to copy you on final versions of contracts and other documents.
One expects a flurry of email to the address to ask for those alternatives.
Some organizations have "central nervous systems" like those of jellyfish: able to react to stimuli (perhaps with a flurry of email and meetings), but incapable of coordinating a conscious plan.
The Obama campaign responded to the burst in fundraising by its Republican rivals with a flurry of email solicitations, including one from the president with the subject line, "I will be outspent".
A small Law School focus group was offering feedback on two different versions of the trailer, one of which would soon be released, and Picker and his production team were exchanging flurries of emails about things like color correction.
While working as a paralegal for several divorce attorneys, I saw firsthand how often each of them zigzagged in and out each week day, leaving a bare bones office staff to deal with urgent phone messages, flurries of emails and over-stuffed client files filled with critical evidence and legal motions that were time sensitive.
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Ha Thuy Vy
MA of Applied Linguistic, Maquarie University, Australia