haven't had my coffee yet ... at first glance i thought you were asking how to make poison-ivy bonzai ...

now there's a challenge.

Ahahahah!

That's about it - just as you say. The longer you leave it (a few years'd be good), the more complex the shape will become, then you can thin it out and probably find that it looks older than the plant actually is.

Should I bother propping it upwards in a "tree" shape now, do it after waiting several years, or will it grow into such on its own in the future?

Thanks for the help!! :D

O.K., if you want a "tree shape", you're going to have to make sure there is at least one upright stem, supported/tied. My own experience with ivy is letting it grow unsupported, spilling out in all directions and forming its own shape that you just wouldn't be able to copy by hand. Bearing that in mind, you could leave it unpropped and have it form a naturally slanting shape.
Whatever, ivy grows so well and so quickly that, providing you have the space, you can try out all of the above and more.
Have fun!

The problem I've seen is that ivy doesn't really develop a tree-like trunk.

An old post I am aware... but I have been working with Ivy bonsai for some time so I would like to set a few poiners to those interested;

With ivy (Hedera) you would want to take a cutting from a matured ivy if you wish to use it as a bonsai.
They have two forms of development to my experience. When they have matured they tend to prove more "woody" and tree like as you would desire. The growth is upward and stronger.

When you take cuttings from a matured plant they preserve these aspects that come with maturity.

Another reason you may want a mature plant as it is (not just a cutting from it this time)would be the fact that as fast and easy as an ivy would grow, it thickens and tapers rather slowly and can take up to decades patience to realise even a shohin size bonsai with a good taper.

Literally ancient trees grown in very difficult terrain tend to have a very twisted trunk (said to be able to better adapt to each root feeding all the branches and also being more flexible to provide resistance to harsh wind or heavy snow).Thus this is a very good reason one would want to go for a Hedera as a bonsai specimen... as it twists and turns in a truly unique way.

But the downfall is the bit about thickening.

With a fresh started plant or cutting, you would best arrange a few sacrifice branches (those you let grow freely a few years and then remove) low and close to the roots to ensure thickening more towards the roots since a Hedera does not easily taper (meaning it more what uniformly thickens through out length of the plant).

I hope this can be of some help

Or you could try young ones and wrap the stems of a few together for a couple of inches and wait for them to fuse (the way ficus do), hoping for an eventual fatter trunk.

im currently trying to make a mame literati style ivy its not much to look at now but the way ive planned it out in about a year it should look great