Are there any known upper bounds on: $$\#\left\{\text{hyperbolic knots }K\subseteq S^3\middle\operatorname{Vol}(S^3\setminus K)<M\right\}$$ ? I expect this grows at least exponentially in $M$, and I am interested to know whether there also exists an exponential upper bound. If we restrict to alternating knots, then work of Lackenby is relevant.

10$\begingroup$ There are infinitely many hyperbolic knots with bounded volume, for example the twist knots. $\endgroup$ – Ian Agol Jan 18 '15 at 8:38

$\begingroup$ Here's a general talk on this topic (without many references): dl.dropboxusercontent.com/u/8592391/Miller%20talk.pdf $\endgroup$ – Ian Agol Jan 19 '15 at 3:39

$\begingroup$ To get bounds on how quickly the above function goes to infinity as M approaches the volume of the Whitehead link complement, the paper of Neumann and Zagier might be of use. $\endgroup$ – Robert Haraway Mar 29 '15 at 18:41
Here is an expansion of Ian's answer.
Dehn surgery on a hyperbolic knot or link generically gives another hyperbolic manifold. This follows from the Dehn surgery theorem; see Theorem 5.8.2 in chapter five of Thurston's book. Furthermore, the new manifold has smaller hyperbolic volume than the original. See Theorem 6.5.6 in chapter six. Some further discussion and references can be found in Ian's answer here.
So: fix a hyperbolic link $L$ in the threesphere with an unknotted component $C$. Surgering $L$ along $C$ gives an infinite family of hyperbolic links, all with volume less than that of $L$. For explicit pictures of this, see the last few pages of chapter five of Thurston's book. (Actually, the pictures go in the opposite direction  he explains how the Whitehead link is the "geometric limit" of a sequence of twist knots.)
This question is so close to two interesting open questions, I thought it might warrant a second answer.
First, as Ian Agol and Sam Need have explained, once the volume $M$ gets above $v_8 \approx 3.66386237671..$ the volume of regular ideal hyperbolic octahedron (aka the volume of the Whitehead link), then there are infinitely many knots in your set. However, it is known that for a fixed volume $v$ the set of hyperbolic manifolds with volume $<v$ can be obtained by filling finitely many hyperbolic manifolds. To give a flavor of this idea, all the twist knots can be obtained by dehn filling one component of the Whitehead link. Furthermore, the Whitehead link is known to be one of the smallest volume (orientable) twocusped manifolds so there can only be finitely many (orientable) onecusped manifold for a fixed $\epsilon >0$, there can be only finitely many hyperbolic knot complements (or more hyperbolic generally orientable hyperbolic manifolds) that have volume at most $v_8 \epsilon$. Finding this number would involve classifying manifolds in this set which do not come from filling the Whitehead link.
Second, one could ask: for a specific volume, how many hyperbolic manifolds have this volume? Hodgson & Masai investigated this question and found examples of closed manifolds that are distinguished by their volume and link complements that share a volume with at least $f(v)$ other manifolds, where $f(v)$ is a function that grows exponentially in $v$. More relevant to your question, Millichamp followed up on this work by considering (amongst a number of other things) lower bounds on how many hyperbolic knot complements have specific given volumes.