Inserts are most commonly used to send the signal from the output of that channel preamp out to some external device (gate, compressor, etc.) and then to return the output of that device back into that same channel on the mixer, thus inserts typically incorporate both send and return unbalanced signals with a shared ground. On the SR32.4 VLZ Pro (which is what I assume you mean by VLZ32), based on Page 15 of the manual (
http://www.mackie.com/pdf/srvlzpro_om.pdf) the channel inserts are 1/4" TRS jacks with Tip as the send, Ring as the return and Shield as the common ground. If you insert a TRS or TS jack all they way in the insert then that breaks the signal path for that channel and if you have no return audio to the insert you will get no audio past that point. That is likely what has happened for you.
Assuming that the Aviom interface is an AN-16/i then with the Mackie SR32.4 you seem to have two options. One option is to use a 1/4" TS jack plugged into the Aviom input put only partially inserted in the Mackie insert so that only the tip makes contact as shown on Page 15 of the Mackie manual. This prevents breaking the channel signal path and might be a simple way to verify that this is the problem but because it is easy for the connector at the mixer to fall out or be pushed in it is not a great long term solution. The better long term option is to connect the Send (Tip) of the Mackie to the Input on the AN-16/i and the Return (Ring) of the Mackie insert to the Thru connection on the Aviom as shown on Pages 23 and 24 of
http://www.aviom.com/library/User-Gu...User-Guide.pdf. You could make your own cables for this but because board inserts are commonly used this way it should be easy to find 1/4" TRS to two 1/4" TS 'Y' cables made for this purpose like these,
http://www.hosatech.com/products/340334/4_in_TS.