The scenario the following:
Can AutoSys and the Web UI remain up and running during this process?
Yes, both AutoSys and the Web UI can remain running during this process.
AutoSys is configured to use its local Oracle Client with the local tnsnames.ora file containing a single Oracle service entry for the 2-node Oracle RAC.
Web UI may be configured to use a local tnsnames.ora file, or it may be configured to simply build an ojdbc connect string. Either way, it uses a single service name for the 2-node Oracle RAC.
The fact that there are 2 nodes is transparent to AutoSys and the Web UI. As long as one of the nodes is reachable using the common Oracle Service name, there will be no issue or interruption. Any connections that are severed when one of the RAC nodes goes down will be reattempted and reestablished on the other node.