Consolidate or something like it

I have a master inventory list. Column C has the inventory type. Column F is the quantity on hand. I would like column G to sum the totals "used" based off of other sheets in the same workbook. The other sheets represent different "booths" that I could use the product at. The sheets consist of a drop down menu system that references the inventory list. Then I manually add the quantity I plan on using for that booth. I can get it to work as long as every sheet or booth has some sort of inventory in it. If the booth is empty (as most are right now) I get a #ref error. I would like to know if it is possible to consolidate all of my booth sheets into the master inventory sheet based on what may or may not be in a set range of cells on each sheet. IE if there is nothing in that sheet range that matches the inventory list it returns a 0 for that item and in other sheets do have something for that item it adds that to the total as well. The inventory on the separate sheet or booths may not be in the same order but will always be in the same column with the quantities in the same column. Possible? I have tried to get vlookup to work and Index/Match and have failed. Thanks - M
Edit- If there is no inventory entered onto a booth sheet I would still like it to work for all of the other sheets that do have something listed on them.

Here's an adaptation to your INVENTORY layout that should work. Once you confirm it works to your liking, you can hide those helper columns added on the right of INVENTORY for referencing you sheets by name.

F10, copied down as needed:

=IF($C10 = "", "", INDEX(Inventory!$C:$E, MATCH($C10, Inventory!$C:$C, 0), MATCH($E10, Inventory!$C$9:$E$9, 0)))