I think that this question can be answered in three parts:
- Are there any exact solutions to any equations of the SM?
- Do we always have to use perturbation methods to solve anything within the SM?
- Are there any exact solutions beyond the SM?
The answer to question 1 is: it depends on what equations you are referring to. For example, the propagators of fermions and photons are exact solutions to the equations of motion of the corresponding free Lagrangian (you cannot say that these are not the parts of SM). Surely, most (almost all) equations of the SM have no exact solutions, such as equations of motion with interactions, Dyson-Schwinger equations, Callan-Symanzik equations, Altarelli-Parisi equations, and so on.
The answer to question 2 is: no! We have many nonperturbative methods to deal with the SM, such as the lattice method, effective field theories, and recent advances on unitary methods. There are also some topological or geometrical aspects which should be treated as nonperturbative, such as anomalies, instantons, solitons, the Batalin-Vilkovsky formalism.
The answer to question 3 is: yes! (1) Many two-dimensional modes can have exact solutions, such as two-dimensional QED, the Thirring model, $d=2$ nonlinear sigma model, and so on. (2) Some supersymmetric gauge theories can have exact solutions, such as the famous Seiberg-Witten equations. (3) Large $N$ expansion allows exact solutuons to many models, such as the Gross-Neveu model, $1+1$-dimensional QCD.
If I have missed some examples or some misundertanding, please help to improve it.