Why don't GameStop, AMC issue shares?

Do they just need time to hire corporate lawyers and investment bankers? Do the corporate lawyers and investment bankers just need time to prepare the documents and find buyers?

Yes to both.

Just as importantly, the existing owners of the company need to vote to dilute their ownership.

Also, when the price is artificially high, it's because of wild speculation inherently based on the existing finite supply of shares. Issuing more would quickly cause the price to collapse. This article on the Beanie Baby Bubble has a good explanation. (There are no "sound bites" to quote.)

One partial answer: some companies are doing this. Bloomberg reported last week:

American Airlines revealed plans on Friday to sell as much as $1.1 billion of stock through what is known as an at-the-market program, which allows companies to issue additional shares at market prices. The announcement came after their stock was up as much as 38% at its peak last week.

Just a couple days earlier, AMC said it had raised more than $300 million through a similar program.

There is a bit of a risk because it does require some setup, and it's possible a company could do a bunch of work setting it up only to find the bubble had burst by the time they were ready to sell. But it's possible.