I'm playing an Artificer and I'm fairly low level still, but I really want to use the spell Animate Objects ASAP. Since Artificers are only half-casters, I'd have to wait until level 17 to do so. I know I could multiclass into Wizard to get it at 9th level, but I am looking for ways to gain the ability to use it sooner. I am willing to homebrew something, but I would like some input first. 

What is you actually want to do with the spell?

Tinkering doesn't animate, but it gives objects unusual properties.

A homunculus can be described as an animated object. So can a steel defender or an artillerist's turret.

A Rope of Climbing, which you can make via infusion, is animated. You could describe several of your other infusions as animated as well although they wouldn't actually function any differently.

Animate more than just 2 or 3 things. Basically have a small army of mechanical bugs.

If you're going to homebrew, you could make an Artificer's version of the Staff of the Woodlands.  Awaken becomes Animate Objects, etc....

Sounds like an easy enough homebrew to make.

Artificers aren't the greatest route to magical power, but you might be interested in the 3rd level spell Tiny Servant from Xanthar's Guide.

Animate Object is a powerful combat ability, but Tiny Servant lets you animate for 8 hours without concentration. At higher levels, you can animate multiple simultaneously with a single casting.

Fastest way? Find or buy a scroll of Animate Objects. As long as the spell is on your list of class spells, you can cast it using a scroll, though there is a chance it will fail and waste the spell.

Scrolls.

Buy them asap.