Is there any reasonably efficient way of performing qudit circuit simulations using Stim? If so, then how much worse would the computational complexity scale?
Stim only speaks qubits, not qudits. All of the supported gates are qubit gates, and all of the internal data structures are for specifically the qubit case.
There are no plans to add native support for qudits to stim.
(So if it's possible it will take the form of mapping qudit stabilizer circuits into qubit stabilizer circuits.)
Since you're mentioning Stim, I guess you mean stabilizer circuits.
I do not know about the capabilities of Stim, but judging from the documentation, it seems to work only for qubits. Nevertheless, the computational complexity is the same, independent of the local dimension of the qudit (assuming that it is prime), i.e. it scales polynomial with the system size $n$. The exponent depends on what you want to simulate precisely, but it is at most $n^3$ (propagating stabilizer states under global Clifford unitaries + stabilizer basis measurements). What can change is the constant in front of the polynomial scalings. This, however, depends highly on how you implement the simulation and is probably not noticeable in high-level implementations in e.g. Python.
Interestingly, some things are even simpler if the dimension is an odd prime, such as the computation of some phases when updating stabilizer states under arbitrary Clifford unitaries or measuring them w.r.t. Pauli observables.