Is there a procedure that needs to be followed in order to get an SKS keyserver to sync with other keyservers (specifically, Ubuntu's keyservers)? If so, what is that process?
If it is a keyserver you manage, it should be a matter of adding peers to the "membership" file, and having the administrators of those peers add your server to their "membership" files. Depending on the counterparty servers, you may find that you need to use PKS synchronisation via email: this is probably best discussed with the counterparty administrators. Specific instructions are available in the SKS README, and if you have a peering relationship with one of the larger keyservers, the admins there can likely help with specifics related to their preferred peering arrangements.
In the specific case of synchronisation with the Ubuntu keyservers, you may either work with other keyserver admins in the wide public set, with which the Ubuntu keyserver regularly synchronises, or to request counterparty configuration directly from the Ubuntu keyserver, open an request at http://rt.ubuntu.com/ . Be aware that the administrators of this service are very busy, so there may be some delay before you get a response. Note that your keyserver may need to meet certain conditions in order to be accepted as a peer.
If you seek synchronisation between some third-party keyserver and the Ubuntu keyserver, you would need to ask the admins of the third-party keyserver to request peering (either to the Ubuntu keyserver or to some keyserver with which the Ubuntu keysrever is peered (directly or indirectly)). In this case, it is probably easier to just upload your key(s) to one of the keyservers already in sync with the Ubuntu keyserver.