Is it possible to obtain a military uniform for a deceased veteran whose family members desire that the veteran be dressed in unifom?

Yes, you should be able to, contact info below be ready to show proof of service with a DD214 or supporting documents.
Marlow White for Army/Navy
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Someone else may have better information but you will likely have to purchase the uniform and awards yourself, I know you can get a one time replacement of awards but I don't know if the next of kin can do that.

If you need to purchase additional awards or uniform items, this site has a good selection.
https://www.usamm.com/

I caution you however, uniforms can become expensive.

You can check with a military surplus store as well. They may have the uniform you are looking for and usually they are relatively inexpensive. Depending on the era of the veteran you may have a hard time finding the type of uniform he/she would of worn.