@Matthew Norman This does not initially appear to be a limitation of the Azure App Service platform, which I am most knowledgeable to speak on. My belief for this reasoning is that the number of connections allowed is normally in the 1,000s depending on the size of your instance.
This is likely a limitation within Oracle that you are hitting. From my research, Oracle seems to impose a limit of around 100 connections. You can increase the limit as described here: https://docs.oracle.com/en/database/oracle/oracle-database/21/jjucp/controlling-pool-size.html#GUID-1909EBD1-E526-4385-9A2A-39C89F985A14
How high you set the limit is ultimately up to you and the strength of your Oracle server. It's also recommended that you ensure your code is optimized to close connections when it is finished with them.
Again, please keep in mind that I am not an Oracle specialist, but I hope this helps get you pointed the right direction. Reply back to this answer if I can assist you further.