Is there an infinite dimensional Lie group associated to the Lie algebra of all vector fields on a manifold?
Morally speaking, the Lie algebra of vector fields is the Lie algebra of $\text{Diff}(M)$, the diffeomorphism group of $M$. The relationship between these is less tight than in the finite-dimensional case: for example,
- The exponential map can fail to be defined at any nonzero time (as mentioned by orangeskid in the comments), and
- Even when defined (say on a compact manifold), the exponential map can fail to be a local diffeomorphism at the identity.
It is also not true that infinite-dimensional Lie algebras are Lie algebras of infinite-dimensional Lie groups in general (see, for example, this MO question), and I have the impression that this is a genuine phenomenon and not just an artifact of working with too restrictive of a notion of infinite-dimensional Lie group, although I don't know much about this.
To add to Qiaochu Yuan's answer, I found out that in synthetic differential geometry (which admits a well-adapted model containing the category of smooth manifolds as a fully subcategory) it is easy to prove that the Lie algebra of vector fields is associated to the diffeomorphism group. Working in such a smooth topos, let $M$ be a space, $R$ the line object, and $D=\{d \in R : d^2 = 0 \}$ the "walking tangent vector".
The tangent bundle is $M^D \to M$ where the projection is the evaluation at zero map. By currying (product-exponential adjunction), a vector field on $M$ is an infinitesimal transformation of the space; more precisely a map $X: D \to M^M$ such that $X(0)=1_M$. In other words, the vector fields $\mathfrak{X}(M)$ on $M$ coincide with the tangent space at the identity to the monoid $M^M$ of endomorphisms of $M$. Moreover, if $M$ is infinitesimally linear, then any vector field $X$ on $M$ satisfies $X(-d) \circ X(d) = 1_M$ and is thus invertible. So we have particular $T_1 (M^M) = T_1 \mathrm{Aut}(M)$.
So $\mathfrak{X}(M)$ is the Lie algebra of both the endomorphism monoid of $M$, and its diffeomorphism group (provided $M$ is infinitesimally linear).
If you're curious about SDG, Kock's excellent introductory text is available online for free.