Putting a microwave in an island?
There are three kinds of microwave ovens:


Countertop - rely on airflow typically on sides, back or bottom
Over-the-cooktop - typically include a fan underneath for cooktop ventilation and may use the same airflow path to also provide ventilation for the microwave oven
Wall Oven - like a typical wall oven, does not rely on any ventilation except in the front.


The over-the-cooktop ovens will probably not work for you. You would be paying for extra features (fan and possibly even a light underneath) and they are just not designed for this setup.

The countertop models may work well. You need to check the specifications. For example, I picked a GE 2.2 cubic foot model (essentially at random) and downloaded the installation manual. That referred to trim kits (as you mentioned) which are specifically designed for mounting specific models in a wall. That is exactly what you need. You are then relying on the manufacturer's design for proper ventilation. In the example I looked at it, it appears that the microwave has bottom vents and the trim kit raises the oven slightly so that it can ventilate properly despite being inside a wall or cabinet. Trim kits can be very expensive, so you may find that a "wall oven" will be cheaper.

The first wall microwave oven I looked at (Frigidaire, again essentially random selection) appears to be a countertop microwave oven packaged together with a trim kit - very similar to the GE in design, with vents on the bottom and the trim kit raising up the oven for proper ventilation.

There is a very wide range of prices (and this isn't a shopping site anyway), but just keep in mind that for safety's sake you must follow the manufacturer's guidelines for installation. Don't try to "roll your own" - not when dealing with appliances that can generate a lot of heat very quickly in a confined space.