The FMCSA rules regarding load securement are confusing and hard to read, but I've read them over many times and can't find anything that says straps cannot be used except when it comes to hauling crushed cars.
Here is my understanding of the laws:
They say that straps must be free from defects, and not contact anything that could cut the strap.
If the straps are DOT rated @ X lbs WLL., and that rating meets or exceeds what is required for the equipment being transported, the straps do not have damage, or come in contact with anything that can cut it, then it is legal.
Any chain can be used to secure a load, but without grade markings on the chain the chain will be figured at the lowest rating for a chain of its size.
I'm not trying to say that straps are better or worse than chain, but as far as I can tell they are not illegal if used properly.
If someone has read something official that is contrary, please let me know. I'm using straps for my skid steer and don't want to get a ticket.