You can use Headspace gas chromatography. Headspace gas chromatography is most suited to the analysis of the very light volatiles in samples that can be efficiently partitioned into the headspace gas volume from the liquid or solid matrix sample.
Headspace GC is used for the analysis of volatile and semi-volatile organics in solid, liquid and gas samples. The popularity of this technique has grown over recent years and has now gained worldwide acceptance for analyses of alcohols in blood and residual solvents in pharmaceutical products.
A headspace sample is normally prepared in a vial containing the sample, the dilution solvent, a matrix modifier and the headspace. Volatile components from complex sample mixtures can be extracted from non-volatile sample components and isolated in the headspace or gas portion of a sample vial. A sample of the gas in the headspace is injected into a GC system for separation of all of the volatile components.
The another possibility (classic, simple) is to isolate of ethanol from chewing gum by hydro distillation and determination of the ethanol directly in water distillate by GC.