can't resist
I've got to make some comments. OP's question was on "ammo" so I'll start there. In hand with that is the "gun" component. Turkey guns and turkey loads have come a long way since I toted my Grandad's ancient (1921) Savage pump on my first turkey hunt. Grandad's gun was a 12 ga, standard 2-3/4 in chamber, with a fixed, full choke barrel, which by todays standards was actually a pretty loose shooting gobbler getter. And it was way too long and a tad heavy with its 28" full vent rib barrel. The old gun did kill turkeys effectively though, with its standard 12 ga 1-1/4 oz of #6 shot, as long as one was working at about 30-35 yds or less. I believe the standard 12ga is still the benchmark and minimally a "full" choke, and ideally one of the turkey full chokes is best.
Specific load arguments abound, but I will suggest minimally a 12 ga standard load throwing about 1-1/2 oz of lead shot as as light as one should go. The "short magnum 1-1/2 oz loads are better. Ideally, the 3"-12ga shell with 1-5/8 up to 2 oz of lead is about as heavy as you need. Yes, you can easily kill a goblbler with less, when everything goes as it should. But when things go less than ideal, a bit more lead and velocity are a good thing. A poorly mounted gun from an awkard position, a thin screen of brush , a juke from the bird at the last split second, a nervy hunter, poor range estimation, can result in a less than ideal shot. Then a bit "more" gun can be appreciated.
The 3.5" 12 ga guns are more,and so is the 10 ga, but I don't think you need them. And the alloy 3.5 pumps by Mossberg, affordable and popular, are vicious kickers. We shoot our Moss 835 with 3" shells, its tolerable, and yields very dead turkeys. The 10 ga in about every version I've ever handled, are heavy, and a pain to run from ridge to ridge with on a wandering bird.
The arguments over shot size are plentiful. I personally think #4 patterns are a bit sparse, at least in all my guns. Most guns shoot #6 well, as there are a good many more pellets per ounce, but you give up some pellet weight and consequently penetration with the smaller shot. That can make a difference on a bird where you botch the range estimation and shoot a bit further than you planned. After shooting #6 shot for a long time, I've settled on lead #5 as the compromise, and have not been disappointed.
I've avoided any conversation on tungsten/hevishot loads as I have no experience with them. At decent turkey ranges, lead kills for me and it's affordable. I ran into a racehorse deal on W-W 12 ga, 3 inch, 1-3/4 , #5 shot a few years back at Wally World ($5 a box) and bought 50 rds or more, and have killed near all my gobblers with that since. My guns shoot that load into patterns as dense as 2 oz loads of #6 shot, so I give up no pattern density, and gain a tad more penetration and bone breaking ability. Till I shoot up what I have, which has been very effective for me, I likely will not switch up.
Put some type of sights on your turkey gun, as all this talk of loads if pointless if the pattern does not land at our point of aim!!!!!!!!!!!!!! The ability to tweak the pattern as necessary with adjustable sights, to have it land where you sent it is important.