Do these look ok for the job? https://www.amazon.com/dp/B07D9J6JLD/ref=cm_sw_r_cp_api_i_pzsEFb5H88PPA
Thanks for the help!
To "drill" the holes, yes. You'll need quick change bits like THIS. Although not designed as a drill, in a pinch, with the proper drill bits, drilling a hole in wood shouldn't be a big deal.
To sink the proper lag screws with their corresponding washers , you also need one of THESE socket adapters and then the appropriate size 3/8" socket (in this example, 9/16"). You should have at least two of these going into a stud, preferably four. And you can buy the screws and washers individually at Home Depot or any hardware store.
Without drilling a pilot hole, I'm not sure if this screwdriver could sink the lag screw. Never tried with a screw this big.
They make light duty 1/4" bits these are just an example.
This will convert any impact or the drill I linked.
Either option keeps you under budget.