http://mdc.mo.gov/discover-nature/outdoor-recreation/how/mushrooms/munchable-morels
It seems that you can hunt in public area.
(ZT)
Where to Find Them
There's good news and bad news about where to find morels. The good news is you can choose from hundreds of thousands of acres of top-notch public morel-hunting land in Missouri. The bad news? Same thing. You've got to narrow the field down to the "right" couple of hundred acres on conservation areas, national forest land and the strips surrounding Missouri's dozen or so big public lakes.
Again, there is an abundance of folk wisdom to help you. The morel-nurturing properties of elm trees are highly touted, as are proximity to apple, ash, basswood and cherry trees.
Hunting morels in recently logged forests is another variation on the theme of finding morels around moribund trees. Roots left in the soil may support a strong bloom of morels for several years. Areas ravaged by forest fires also can produce prodigious morel crops. it is unclear whether this is due to the abundance of dead trees, increased penetration of sunlight, or both, but one burned forest tract in Austria reportedly produced 44,000 pounds of morels in a single season.
Aside from these general tips, morel hunting is a matter of wandering about more or less aimlessly, while scanning the forest floor for a glimpse of the spongelike surface of your quarry. When you spy one, don't rush right over and pick it; you may trample a half-dozen others. Morels rarely grow singly. Where you find one, you'll usually find at least one or two more—sometimes a lot more.