What is the test called to test for glutamate sensitivity or high levels of glutamate? Can a gp order this?

Gluten . Not sure if you mean Gluten Sensitivity !! Glutamate is different from Gluten . Coeliac serology can be ordered by your gp for Gluten sensitivity. .
Glutamate allergy. Msg reactions are not an allergy but can cause a syndrome including headache and facial and chest pressure which can be distressing but is self limited. While foods with artificial monosodium glutamate are labeled, many foods contain enough naturally produced glutamates to cause symptoms. These include yeast extracts, hydrolyzed protein or even natural flavorings. I know of no blood test.
Gliadin antigen. The test for gluten sensitivity is gliadin antigen and it can be ordered by your gp. You want to just try a gluten free diet and see how things go.