Accumulators are easier to produce. Volta's battery design could be complemented by a manual worker hot-swapping batteries to the electric engine, just in the same way in which a manual worker need to shovel coal inside the furnace for a steam engine to run.
These are harder. Mostly because the transfer of mechanical energy to electric energy is a "recent" discovery. However, note that such discovery does not depend on the invention of a steam engine.
The answer is water turbines.
Mills, be it watermills or windmills or horsemills, have existed for a very long time. Some claim that Egyptians and Babylonians have been using them too.
Gears too have existed for quite some time, with obvious implications that one could couple a watermill with a gearbox to increase the rotational speed derived from simply splashing water on the turbine.
Finally, plug that to an alternator, and enjoy some good wattage. Fun fact: magnet-based alternators work both as electricity generating devices, as well as electric engines.
If large amounts of power are not a must, any SE user could build that in their kitchen sink.
--- EDIT for science-fiction aficionados, and not for long-lasting factory work.
Could some form of electric engine be invented with the knowledge from the XVII and XVIII centuries?
Galvani and the Frankenstein's propulsion
Galvani's experiments on dead frogs showed that it is possible to activate muscles using electrostatic potentials. Previous studies on anatomy had made it clear that limbs move thanks to the action of pairs of antagonist muscles.
The Frankenstein's propulsion works by using parts of large dead animals. For instance, the shark-propulsion uses dead sharks attached to the back, or the bottom, of a boat. Electrodes are inserted in the dead body, with the terminals in the desired muscle groups, and sealed with gum and wax. A simple hand operated mechanism, such as a wheel with a rotating lever, rhythmically discharges a Volta's pile via the electrodes, causing the antagonist muscles to contract asynchronously, hence generating movement. For instance, this could be used to make the dead shark flap its tail.
The Frankenstein's engine lasts as long as the battery lasts and as long as the muscle groups don't start rotting. Injecting the veins of the animals with adequate preserving fluids may lengthen the life of your Frankenstein's engine.