。
Supplies are starting to run out.
Literal
Supply [subject-が] run-out-begin is.
尽きる ('to run out, be exhausted') + ~始める ('begin to') forms the compound verb 尽きはじめる. Compound verbs of the form [verb stem]+[verb] are productive in Japanese: 食べ始める (begin eating), 歩き続ける (keep walking), 読み終わる (finish reading). The ~ている wraps the whole thing in an ongoing-state reading.