I agree with David Siegel's answer but I think its always important in the law to consider what it would take to get what you want. I will say it is possible but not as an export prohibition per se and practically willmay fail.
The state could use its 5th amendment powers of eminent domain to take possession of all of the desired raw materials and relevant mineral rights for just compensation thus making them all property of The State. Expensive and not sure what the incentive is, but as laws are now, permissible if there is some public benefit. At this point the relevant natural resources would be property of the state and could therefore control where it goes such as keeping the resources in state.
Now assuming that neither budgetary restrictions nor a lawsuit were successful at stopping this measure (two big ifs), The federal government could still take the resources from The State in the same way under eminent domain.
In summary it is possible if:
- Your state has large excesses of cash and a desire to be an irrational economic actor AND
- the federal government is shutdown for a very very long time preventing operations of the courts and/or solicitor generals office.