I have submitted a package to CRAN and am interested in how much it has been used. Is there a way to get the number of downloads from the CRAN website?
No. Because CRAN has many mirrors that appear and disappear it has generally been the view that these statistics cannot be collected in a reliable and satisfactory way -- archives of r-help and / or r-devel will have prior discussions.
Some people have started to stick their heads together to change this, but as far as I know have no visible results either.
The crantastic site tries to collect community-driven reviews etc which 'eventually' would act as a signal for use, popularity, quality, ... but it is not quite there yet either.
As Dirk says, there's no way to get download statistics. This won't be relevant for your particular package since you just submitted it, but in the long run, here's one convoluted approach for determining your package's "usage":
If your package is general enough (ie. might get re-used by another package), you can easily apply social network analysis to look at the centrality (e.g. by closeness and betweenness) of your package to other packages by using the packdep package. This will return an igraph object, which can be analyzed in many different ways. I did this for a talk on R in Finance a while ago, and it seemed like an effective way to determine the importance of general packages. Here's a simple example:
library(packdep)
d = map.depends()
centrality(d, order.by = "betweenness")
This is similar to studies where people study academic papers' references to determine the "importance" of a given paper.