Regarding partitions of unity: Every locally compact group is paracompact, thus it is normal, thus there exist partitions of unity. But you do not need this argument: For general locally compact groups and any compact subset $A$ and any finite precompact open cover of $A$ there exists a continuous function with the value $1$ on $A$ and vanishing outside the union of the cover which can be written as the sum of continuous functions supported on the elements of the cover (Folland, Real Analysis, page 134). You use the cover described in the comment by Lee and then you take these functions given by the theorem and multiply them with the function $f$ (the proof uses the Alexandrov compactification which is compact, thus it is normal).
First I thought you were only interested in an approximation (because you had mentioned Stone Weierstraß), for an approximation you can use this proof:
If $\mathcal{U}$ is a local basis at identity consisting of compact sets and $(f_U)_{U\in \mathcal{U}}$ is any family of continuous, non-negative, symmetric functions such that $\int_G f_U=1$ and $\mathrm{supp}(f_U)\subset U$ for all $U\in \mathcal{U}$, then $(f_U)_{U\in \mathcal{U}}$ is an approximate identity for the convolution algebra of compactly supported continuous functions (reference: Gerald B. Folland, A Course in Abstract Harmonic Analysis, proposition 2.42). Since convolutions can be approximated by finite linear combinations of left-translates of the functions, we get your result.