Is it theoretically possible to artificially create amber?
Yes, and this has been done since at least the mid 19th century (Răducanu, 2006) initially it was done to keep up with the demand, but now is a source of scrutiny for valuation of samples (Pederson and Williams, 2011; Răducanu, 2006).

Imitations have been produced by heating the following (Răducanu, 2006):


Polymers, such as polyester and polystyrene
Casein
Glass
Celluloid
Other resins, such as phenolic resin
Copal - this is referred to as 'pre-amber tree resin', a natural intermediary substance between tree resin and amber.


Both Răducanu, 2006 and Pederson and Williams list several techniques as to how to tell them apart from authentic amber.  Pederson and Williams, 2011, also detail that some artificial samples are so close to amber that spectroscopy techniques are required to tell them apart.

References

Pederson and Williams, 2011, Copal vs. amber, Organics

Răducanu, 2006, Actual Exigencies Concerning the Quality of Amber
Pieces Commercialized in Romania, Universităţii Petrol – Gaze din Ploieşti