Incremental upgrades - Is it required?

I have a new client who is running 1.3.8a and was told to upgrade because of an incompatibility with the newest USPS shipping module. I am going to be taking care of the update from my client. Is it necessary to do incremental upgrades all the way up from 1.3.8a to 1.5.1 ? Or can I take the jump from 1.3.8a to 1.5.1 ?

You can effectively go straight to v1.5.1. The database upgrade will automatically step through every intervening version. There may be some customizations that have to be redone because of differences between v1.3.8a and v1.5.1, and some mods have not been upgraded and will need to be scrapped or worked around - whether this is an issue for you depends on what mods and customization you have.
Just because a mod's description doesn't say "v1.5.1" does not mean it is incompatible; many didn't need to be changed for the new version.