Can I mount a ring flash to the kit lens of the Olympus OM-D E-M10 Mark II?

No, on this particular lens. The ring flash would use the filter thread which is positioned around the front lens. The ring flash would add weight on the tip of the lens. Since the lens is a small motorized zoom lens, I would not trust it to support a ring flash. The motor and zooming mechanism risks premature breakage, even if not activated during the use of the ring flash. I would use a ring flash on a prime lens or on a sturdier zoom lens, mechanically speaking.

On flash compatibility

A flash in manual mode (no TTL, no HSS) can work on any camera as it only uses the center pin to trip the flash. However, if you want TTL/HSS compatibility make sure the flash socket is made for Olympus OM-D E-M camera. A manual flash won't be a problem for most situations. See this topic covering the problem in much more depth.

My advice

As for beginning into flash photography, i would advise to choose an off-camera system (wired or wireless). And experiment with various flash modifiers. Bare flash light is not always a desirable light. Cheap flash modifiers can be made for a couple of $€. Free DIY plans are available all over the web. If you are not into DIY, it will be more expensive (sometimes more than a new flash!). What you want to fiddle with : bounce (cards or walls), honeycomb grid (of various angles), diffuser (various forms), beauty dish, barn doors, snoot + honeycomb... You can even combine them, sky is the limit !

The advantage of this strobist style is that you can dissociate the camera and flash placement relative to your subject. Also, you can change the type of lighting you want on the cheap with flash modifiers.

If you wan to stay on ring flash, stick with xenon tube flash (avoid led) for better light quality. To widen your search, you might also consider twin flash systems. They produce similar lighting conditions than a ring light. The power of a flash is determined by its Guide Number (GN). For a better usability, try to get a flash with a Guide Number around GN60 or more. Lesser GN are more targeted at macro and proxi photo.

My opinion on brands

Yongnuo is one of the cheapest brand of flash. Expect sample variations in lifespan and aging color cast after some years. However, it can be a good source for trying out something new on the cheap and choose a better flash system later. They have a ring flash YN14EX but only for Canon it seems. Maybe other makes are supported on the new version II.

I own Godox. For me, Godox is the best brand for cheap but reliable and qualitative flash system. I use a v685 flash with a X1-Pro wireless transmitter. The AKR1 modifier kit + S1R adapter allows me to have some magnetic modifiers ready to use under a minute. However, I also do some other modifiers in DIY fashion. In some situation, I would like to have a second v685. I might buy another one in the future to complete my kit or an AD200. They have a few ring flashes (a big one for studio: AR400, a normal one: ML150).

Nissin delivers compact but powerful flashes. Best sized for smaller mirrorless cameras. However, that comes with a price (2 to 3 times Godox price). They have good wireless controllers with physical dials (best if you want to set it without looking). Their ring flash (https://www.nissindigital.com/mf18).

Paul C. Buff make pro grade lighting equipment that not so costly. Their ring flash Alienbees ABR800 is dope with a hand grip and a strap. I would buy one if I have to get pro.