"obligated to assist"? An unfortunate turn of phrase. Either they want your business or they don't. That said, getting housing for pets is several orders of magnitude higher difficulty than just yourself (highly dependent on location). Pets do damage housing one way or the other (the other trying to put someone with allergies in that housing next). Expect to pay large and probably non-refundable pet deposits or take sub standard housing. Your agency will not cover those additional fees (everything comes out of one pot anyway so any additional expense will always lower the rest of your pay). Large dogs over 40 pounds such as German Shepherds will be extra difficult as that is often the dividing line between a property manager accepting a pet or not.
It certainly can be done and lots of travelers bring pets, even large ones. Just expect that it will be more difficult depending on location and more costly. Your agency has only so much in the way of resources to find property for your and they are often reluctant to sign for housing they haven't used before because of the much higher risk of rejection by the traveler. Often you will be able to find housing much more readily than your agency, especially with the special needs of three pets. Going the RV route is a great problem solver as parks are usually very pet friendly. Of course, that requires a substantial commitment to equipment and I don't blame you for being reluctant. Some assignments are not RV friendly as well, that is to say you may not be able to find a park with 30 miles or so.
Good luck! Woof!