The mobile emitter came from a ship that was 6 meters in length and only had a single crewmember. Why would it have a holographic emitter on board?
From the Memory Alpha Article on the mobile emitter:
Owned by Henry Starling in 1996, the emitter was either aboard the appropriated Aeon timeship, or derived from the 29th-century technology therein. Starling used the emitter to grant mobility to his captive, The Doctor, while the latter was displaced in time after he hacked into Voyager's computers and stole the hologram, as equipping The Doctor with the emitter gave Starling the ability to use him as a hostage by threatening to destroy the emitter as it held The Doctor's program. After being returned to the 24th century by Captain Braxton, the USS Voyager was allowed to retain the anachronistic mobile emitter, granting The Doctor a new-found mobility and utility beyond Voyager's sickbay and holodecks, able to slip the emitter on and off relatively easily.
So, it is not clarified where the mobile emitter comes from. I personally prefer the aboard timeship theory. The reason is the following:
From the artilce on Futures End, Part 1
Braxton explains that he was trapped on Earth thirty years ago, in 1967, when his timeship crashed in the mountains. He initiated an emergency beam out but was unable to reach his ship in time and instead it was found by Henry Starling, who began exploiting its 29th century technology, beginning the microcomputer revolution of the 20th century on Earth. Braxton believes it's too late and that "the future's end" is inevitable. He explains the paradox he believes will occur.
So, in the 24th and 29th century, they are still aware of the microcomputer revolution that began in the 20th century with Starling. So, he designed products that he was able to sell. Would the mobile emitter have been something he was able to design, Voyager would've known the design and be able to reproduce it. So, IMHO, Starling has not been able to reproduce the Mobile Emitter. (I am aware that the design could've been forgotten during WW3)
Now, why does the Timeship have a mobile emitter on board. Assume you travel alone around in time. At some point, you might run into a problem and no way to reach anybody from your timeframe. The mobile emitter is an easy an convenient way to have all sorts of specialists (doctor, engineer, hunter (should you crash and need food) and so on) at your disposal at all time. So, it makes perfect sense for a ship to have such a piece of equipment with them.
The Star Trek Book describes the Mobile Emitter as being a piece of technology, from the 29th Century ship, that Starling appropriated. This confirms that he found it rather than having built it himself.
Starling equipped the EMH with a stolen autonomous self-sustaining mobile holoemitter, technology unknown to the 24th century.
Since it's compatible with the Voyager EMH, it seems likely that the ship had its own EMH.