score:8
No. Status miles (including both MQM and MQD) may only be earned by the person traveling, not the person booking/paying.
If you enter your SkyMiles number for the booking, then nobody will earn miles as the name on the booking will not match the name on the SkyMiles account.
This is true for basically all Frequent Flyer program for all airlines. There are a few potential exceptions (eg, a small number of airlines allow pooling of points between family members), but these are very much the exception, and not relevant for Delta.
Upvote:6
What Doc said, but I'll add that Delta does have a SkyBonus program for small businesses where the purchaser of travel for others earns a different type of points that can be redeemed for free travel and other benefits, including SkyMiles Silver Medallion status.
But the rules for this program, and similar ones offered by other airlines, are very much intended to restrict its use to businesses that buy significant amounts of travel for 5+ employees, not a family or personal use, and the benefits mainly accrue to businesses spending tens, if not hundreds of thousands, of dollars a year with Delta, so it's no help for your situation.
Delta does have some ways to get MQMs and MQD waivers through their credit card programs, though whether any of those make sense for you or whether it's worthwhile to get to silver is a different question depending on your situationa nd future plans.