We report on electron wakefield acceleration in the resonant bubble regime with few-millijoule near-single-cycle laser pulses at a kilohertz repetition rate. Using very tight focusing of the laser pulse in conjunction with microscale supersonic gas jets, we demonstrate a stable relativistic electron source with a high charge per pulse up to 24 pC/shot. The corresponding average current is 24 nA, making this kilohertz electron source useful for various applications.