I think this is a good idea, but $5,000 to 1,000 people would probably be even better.
The country has already suffered enormous financial loss thanks to investments in tracking, emergency medical support, vaccine development and distribution, and economic stimulus. That's on top of the massive economic damage suffered by lockdown, stay-home campaigns, and ongoing restrictions. A $1k payout per citizen for getting vaccinated would be a drop in the bucket by comparison - especially if it means finishing off the pandemic.
EDIT: Is anyone downvoting willing to explain why? I'm genuinely curious.