What alternatives do I have to make this rocking chair usable once again?
Obviously the full options include re-doing the seating with some kind of weaving. This can be a DIY thing, but really it's a job for a professional unless one is really dedicated.
The simplest solution I can think of is something that anyone could tackle and that is to inset pieces of plywood cut to a close fit to the existing rebates/rabbets on the seat and back. A bit ugly perhaps but serviceable. Adding a chamfer or roundover to the front edges of the plywood will improve the appearance a little and help reduce the chance of the edges becoming damaged over time.
Both chamfering and rounding-over can be done using careful sanding only if you don't own a router or other suitable tool for doing chamfers or roundovers.
Making these plywood panels accurately to shape could be challenging but will be made much easier if you begin by making stiff card or corrugated cardboard templates, which can be cut easily with scissors and knife until the fit it just right (hint, corrugated cardboard from boxes cuts very well with a breadknife used as a saw). Then you simply trace around them on to your plywood and cut it out.
Basic upholstered panels
One step up from this is to use plywood panels as the basis for upholstered panels as per @FreeMan's Comment, with foam and/or other soft padding and a tightly-stretched cloth cover. Note that the plywood needs to be reduced in size all around to allow for the thickness of the cloth, especially at corners where folds/pleats need to be done.
It's beyond the scope of an Answer here to describe upholstery techniques in necessary detail so further research will be needed if you choose to go down this route.