Every program in this world has a point at which you have to say, "I can't do anything about it". And this is how you say this:
Try
.....
Catch
''Do Nothing
End Try
So, you may want to write to event log about your special logging failure. But if event log fails, you can't do nothing about, and the code above will take care of that.
In your concrete example, you should use a lock in your method because multiple threads may access same code and the file may be locked, which will cause the error.
SyncLock lockObject
'' your code
End SyncLock