They don't list which specific Epson/Canon printers they use, so I'm assuming then that they will properly convert the image for me depending on which printer they choose to use for the print.
Some labs will share the ICC profile for the printer they are using.
Even Costco used to do this!
Ask the lab if they offer that, and where you can access them.
If you are able to access them, then you can (with a lot of digital post processing software) download that ICC profile to the appropriate location on your computer and then link to that profile in the "proofing" part of the Colour Management portion of your digital post processing software.
If you print to a number of different printers, you can choose the correct profile for the targeted printer.
You, can then, in the proofing mode, toggle back and forth between the generic profile you do most of your work in, and the ICC profile that yields a more accurate representation of how the print will actually look.
If you do lots of printing at that lab, it may help to automate adjustments that tend to correct for that printer's characteristics.