A company's share price typically has 2 impacts on its internal operations:
(1) If it needs additional funding to accomplish its goals [like expanding to new locations], then it could do that by (i) waiting for accumulation of net earnings to fund expansion; (ii) take on more debt; or (iii) offer additional equity [like more shares offered to the market].
When offering new equity, consider - offering an extra 20% of additional shares decreases the proportionate ownership of the old shareholders, but the extra money coming in makes the pie bigger. In theory, offering new stock should basically keep old shareholders at the same amount of value owned [omitting transaction costs and volatility of hype around new projects]. However, if the company's management believes that the stock market reflects a very high share price relative to what management believes it should be worth, then it might be able to 'cash in' on that by offering shares for this funding.
So if the GME price increase lasted long enough to be acted on by internal management, they could put out a share offering at, say, $100 / share, let's assume doubling the total shares outstanding. If the old share value of ~$20 / share was a more realistic evaluation of the company's future profits, then GME basically added value to old shareholders, who would be getting a pie that would be about 2.5x bigger [$20 / share vs $100 / share, for the new 'half' of the company's equity], and only losing 50%, increasing the value the old shareholders hold. So, this would be acting in the interest of current shareholders if it was possible*.
*Note that this isn't likely possible, as the bubble is likely to burst on GME's current run within maybe days or weeks, and it could take months to get everything prepared to do a share offering. Keep in mind also, that GME has lost money every quarter from its operations, over the last 2 years. So an influx of cash might not even help it survive.
(2) If a company has an employee stock option program or similar, then a rising share price might change the incentives of key individuals to stick around. Of course, if it seems like a big bubble and an employee is legally able to sell, this also might create a risk that such an employee would just 'cash out' and leave.
In short, share price movements can impact the internal workings of a company, but this high level of volatility that is likely to end quickly doesn't have a big predictable outcome.