Yes you can buy Buck Deer and Spike Elk tags over the counter.
But anterless tags are a draw.
Most Southern Utah deer tags are on a lottery systems. There are never any tags left over after the draw.
Some Northern Utah tags can be bought over the counter, But most units they sell out long before the hunt.

If you think you can show up and buy tags the day before the hunt, You will be very sorry.

70% of Utah is public lands. So access isn't so much a problem as knowing where to go. Elk in particular move away from the roads and hunting pressure as the season opens.
Most private land is now run as a CWMU where land owners sell tags at high prices.