In general, the laser centers itself on the lines you draw in your file and usually burns away about 0.005" of material from either side of the line. With polyethylene (PE), the laser might take off closer to 0.02" of material from either side of the line for sheets around 0.05" thick (possibly even more for thicker sheets).
The cuts in PE usually have a lip or ridge to them, and very long, thin strips in a part will melt or warp. Thin, sharp, acute angles also tend to melt and warp. If you care only about the holes that are left in the plastic (and not the pieces that fall out), then even though the part itself might come out gnarled or warped, the hole left in the material might look acceptable.
If you like, you may email us your design and we can let you know if laser cutting it from PE would work.
If a different thickness of material is acceptable, then johnsonplastics.com has 1/32" acrylic-based, adhesive-backed plastic that comes in a variety of colors. You can see them by following these links in their online catalog:
Engraving Products / Plastic / Rowmark / ADA Alternative / 1/32" (second 1/32" option in the list; part numbers end in “A” to indicate adhesive backing)
They can also apply an adhesive on other materials for you, in which case you can look at their 1/16" and 1/8" offerings as well (you would have to call them for pricing and lead time). Note that you would need to select a laser-safe material (look for a thunderbolt icon at the top of the catalog page).