Eventually? Yes. In the near term? Probably more as a supplement. The biggest net gain for forecast modeling is the synoptic flights by the gulf stream. The P3 and C-130 flights (and I know I'm just using the short designator and not the full model ID) are more to tell you what's actually going on in the storm at this very minute. NOAA/NWS wishes that they could fly the gulfstream a lot more often than they do. The reason being is that the atmospheric measurements that it gives are fed directly into the computer models, giving them much better starting points for their computational modeling runs to predict the movement of the storm. Right now, they get a gulfstream flight maybe once every 24 hours at best. Usually, its a longer turnaround from what I remember reading (granted, that could have been shortened aince it's been years since I read up on this).
Where a drone might be able to help is in being able to repeatedly run long duration sampling missions around the surrounding environment of the storm. You could have a few of them on a continual cycle that are constantly feeding up to date information into the computer models. A limiting factor will be the dropsonde supplies. Since most drones are much smaller than the Gulfstream and P3/C-130, they won't have the magazine capacity to drop a lot of those sensors into the air. With some of the larger platforms however, canister launchers could be installed where existing hardpoints are for ordnance. That could cover a lot of the need right there.
As for replacing the C-130 and P3 for storm penetration duties, that's a different animal. Smaller drones would have significant issues with handling the turbulence aloft in those storms. Larger platforms are CURRENTLY more optimized for loitering in "still" air. That isn't to say that they can't be made, just that it isn't economical to do so. The existing frames used for that duty are either second hand from the navy, or are just repurposed frames that were designed for other uses. There have been attempts at purpose built smaller platforms for storm chasing duties in the past, but, they never proved to be economically viable. You might see something when a optionally piloted large transport is developed in a few decades.
Generally, the storm chasing endeavor is an expensive one already. But it doesn't have the budget to develop bespoke platforms or to even purchase brand new air frames. It's going to be second hand drones that the military and DHS is done using that will eventually filter into NOAA/NWS use.