Yes you can, but they are a hard "get"
First off, if you are in weeks, understand how many TPU you will need for the time period you want to stay, there is a trading power calculator on the RCI site that can help with this. Disney resorts are generally a fairly high TPU value, so you may need to combine a couple of deposits to get sufficient trading power. I actually generally keep a combined "super trader" deposit in RCI to use for Ongoing Searches (OGS). I do this by gathering up an expiring deposit and all of my change deposits at one time and combining, which also extends. My current super trader is worth 79 TPU so will match most anything.
If you have a sufficiently powerful trader start an OGS for the resorts you would like to stay at in the timeframe you want to stay. This should be done well in advance, and you will have to pay the reservation fee at this time. Note that Disney deposits rarely make it to "public" sightings as they will be used to fulfill OGS first. Also, if you upgrade your membership level in RCI, you can boost your match level too; I believe Platinum members get first dibs on OGS (Priority Access); you also won't have to pay the OGS reservation fee in advance.
If you are RCI points, some of the above applies, but I don't know the ins-and-outs of that program as well. TUG (Timeshare User Group) is a great place to get educated on all the details of trading into Disney, or anywhere else for that matter.
Is it a pain to trade into Disney? Sure. Is it a sure thing you will get it? Absolutely NOT (have a plan B in place). Is it economical? Absolutely; I can pull together about $1500 worth of TPU plus fees to be able to get a week at a Disney resort. It's not easy, and I don't bother because we love Vistana Resort which is near enough to Disney for us, much less $$, much less hassle, same great quality.