We sell what we call "zero phos" fertilzers all the time in those markets that are having Phosphorous related troubles. There aren't really any agronomic problems you can expect, though you should emphasize the need for correct soil Ph to insure what P is there is also available to plants. Companies that sell fertilizer should respond with a good selection of no-phos blends that will fit your program. Phos can cause more problems with weeds & you may enjoy the removal in the long term. Summer anual weeds enjoy the surplus P that many applicators use. Creeping & vining weeds typically benefit from surplus P too.
Also, some preemegrgent herbicides are known to have trouble "coming off" the P when used in fertilizer combination products.
The trouble with P is one of solubility. P is actual too insoluble. The most readily available & therefore most desirable P source is DAP or diammonium phosphate. The old Triple Super Phosphate could take 200 years to enter the solution entirely. As soluble as DAP is, it will still take 50 years to go entirely into the soil solution. So most applied P has the chance to move laterally during large scale rain events. This runoff is know to enter some of our nations waterways, especially from sheep farming, I'm told. Combined with other domestic, agricultural, & industrial phosphates, our industry needs to do it's part to help improve the nations waters. I think you'll find that it's not hard for us to do without applied P.