Is it actually using syslog, or writing its own log files?
If it is using syslog, then you just need to be able to talk to
/dev/log (which is generally a symlink, so probably also the target of that link). It’d then be up to the daemon listening on that socket to decide where to write the log files (or even be configured to send the logs to another system over the network). I don’t think there is an interface that allows this yet.
If the process is running as a daemon under systemd though, it should already be able to write to the system log by simply writing to stderr though.
If it is writing the log files itself, then presumably it could be modified to write them to some other location, or be modified to use one of the above options?