Nice answer BS. I was about to post something similar but I didn't have a proof of the non-linearity of the action of $k$-jets of diffeomorphisms. One additional remark is that you can put on a vector space structure if you have additional structure on $M$. Sufficient would be to choose at each $x \in M$ the $k$-jet of a diffeomorphism from $M$ to $T_xM$ sending $x$ to $0 \in T_xM$. For example if $M$ is Riemannian the $k$-jet of the inverse of the exponential map would do or if $M$ is a submanifold orthogonal projection onto the tangent subspace would work. In such a case you set up a bijection $$ J^k_0({\mathbb R}^n, M)_x \to J^k_0({\mathbb R}^n, T_xM)_0 $$ and the latter space is a vector space because $T_xM$ is a vector space.

Nice answer BS. I was about to post something similar but I didn't have a proof of the non-linearity of the action of $k$-jets of diffeomorphisms. One additional remark that might help the OP is that you can put a vector space structure on $J^k_0({\mathbb R}^n, M)_x$ if you have additional structure on $M$. Basically you need to be able to determine a family of co-ordinates (actually $k$-jets of co-ordinates) that are related by linear transformations to avoid the non-linear action of diffeomorphisms when you change co-ordinates. Also you want to choose different co-ordinates at each point of $M$. Sufficient would be to choose at each $x \in M$ the $k$-jet of a diffeomorphism from $M$ to $T_xM$ sending $x$ to $0 \in T_xM$. For example if $M$ is Riemannian the $k$-jet of the inverse of the exponential map would do or if $M$ is a submanifold orthogonal projection onto the tangent subspace would work. In such a case composition with the chosen $k$-jet of a diffeomorphism defines a bijection $$ J^k_0({\mathbb R}^n, M)_x \to J^k_0({\mathbb R}^n, T_xM)_0 $$ and the latter space is a vector space because $T_xM$ is a vector space.