Yes – weight (2, 2) (often simply called “weight 2’’) Siegel modular forms of genus 2 have been, and are still being, computed, but not by the same cohomological / modular–symbol method that works so well for weight 2 elliptic modular forms.
The successful calculations use a mixture of

• explicit Fourier–Jacobi expansions,
• Jacobi-form and Gritsenko-lift technology,
• restriction to Humbert surfaces,
• point counting on moduli spaces, and
• ad-hoc linear algebra,

rather than the Selberg trace formula.

Below is a guide to what has actually been implemented and where the resulting tables can be found.

1. Level 1 : nothing to compute
   For the full Siegel modular group Γ₂ = Sp₄(ℤ) the space S₂(Γ₂) of weight-2 cusp forms is 0. (This goes back to Igusa.) Hence all interesting weight-2 forms necessarily come with level.

2. Paramodular level K(N) (conductor N of an abelian surface)
   This is the case predicted by the “paramodular conjecture’’ of Brumer–Kramer: a simple principally polarised abelian surface A/ℚ of conductor N should correspond to a weight-2 paramodular newform of level N.

a. Dimension formulas
– T. Ibukiyama, “Dimension formulas of Siegel modular forms of weight 2 and supersingular loci’’ (manuscripta math. 1992).
– H. Katsurada, programs (Fortran/Magma) implementing Ibukiyama’s formulas, give the dimensions for every N (especially prime N) and show that weight-2 forms exist already for quite small levels.

b. Explicit Fourier expansions, Hecke action, tables
– C. Poor & D. S. Yuen developed Maple / Magma code that constructs the whole space S₂(K(N)) for many N by writing a paramodular form as a finite Fourier–Jacobi expansion and then imposing the functional equations coming from the group.
References:
• “Paramodular cusp forms” (Math. Comp. 84 (2015), 1401-1438) – prime levels N ≤ 997.
• “Dimensions of cusp forms for Γ₀(p) in degree 2 and small weights’’ (Abh. Math. Sem. Hamburg 83 (2013), 87-111).
• Data files are available from Yuen’s website and have been imported into the LMFDB (https://www.lmfdb.org/ModularForm/GSp4Q/).

   – These tables contain, for each newform, sufficiently many Fourier coefficients to determine the eigenvalues of T(p) for all p ≤ about 200 and to reconstruct the spin L-function numerically.

   – Brumer–Kramer, “Paramodular abelian varieties of odd conductor’’ (Trans. AMS 366 (2014), 2463-2516), used Poor–Yuen’s forms to match all simple abelian surfaces of conductor N ≤ 1000 that were known to them (conductors 277, 353, 587, 713, 731, …).

c. Implementations available to users
– Magma 2.19+ (F. Cléry, G. van der Geer) contains a package SiegelModularFormRing which, for genus 2, can construct weight-2 forms of paramodular style and compute Hecke operators.
– SageMath has bindings to the Magma code; independently, D. Yasaki wrote Cython routines for weight 2 at square-free level.

3. Congruence subgroups of Sp₄(ℤ) (Γ₀(N), Γ₁(N), …)
   Much less has been tabulated, but isolated computations exist.

– Ibukiyama & Katsurada give dimension formulas for Γ₀(p).
– Poor–Yuen’s code works for Γ₀(N) as well, and they have calculated many prime levels p < 100, though most of these spaces are generated by Gritsenko lifts.

4. Cohomological / point-count approach
   Although weight (2, 2) is not cohomological, one can still obtain Hecke traces by counting points of Siegel moduli spaces over 𝔽_q and applying the Lefschetz trace formula (in spirit similar to what the OP suggested).

– C. Bergström, C. Faber & G. van der Geer, “Siegel modular forms of degree two and three and counts of curves over finite fields’’ (Selecta Math. 19 (2013), 987-1013).
They compute the trace of T(p) on many local systems; after subtracting Eisenstein and endoscopic contributions one can isolate the non-lift forms of weight 2 in level 1 and small level, confirming that none exist at full level and giving partial traces at small levels.

The method is powerful conceptually but, so far, has not been turned into a routine tabulation tool for arbitrary level.

5. Trace-formula calculations
   Serious analytic trace-formula work for GSp₄ (Luo, Sarnak, Weissauer, Pitale-Saha-Schmidt, …) gives averages of Hecke eigenvalues but has not yet been implemented to yield explicit bases of weight-2 forms for concrete levels. All published tables come from the explicit-Fourier-expansion methods discussed above.

Summary.
• For level 1 there are no weight-(2,2) cusp forms.
• For paramodular level K(N) there is a complete tabulation for square-free N ≤ 997 (Poor–Yuen), and Magma / Sage can reproduce and extend these results.
• These tables have already been matched with abelian surfaces by Brumer–Kramer and are incorporated in the LMFDB.
• No comparable data set has been produced with the global trace formula; the practical computations rely instead on Fourier–Jacobi expansions and on the fact that weight 2 is small enough for linear algebra to terminate quickly.