How do I place sequential serial numbers on images for export?

Yes. In the Export dialog box, under "File Naming"  check "Rename To" select "Custom Name -- Sequence" then type in your custom name and it will automatically add the sequence numbers to the files. I'd suggest you export these to a "specific folder."

I don't think Rusty's recommendation will place the image name on the image. However, you can using print to file in the Print module. By checking the "Photo Info" option, then from the drop down options to the right, you can select Custom Name, or several other options of metadata to print on the image directly.

Additionally you can take a look at the Mogrify export plugin ... it can do this via the Export process ... it is donationware so test it out before you donate in case it is not quite what you expected.

http://photographers-toolbox.com/products/lrmogrify2.php

Then, let the OP clear it up: Does he want a watermark with the file name and sequence number of it, or a filename that includes a sequence number.

Frankly, creating a filename with a sequence number for each image would be alot easier; you can do it in the print module, and you still have to create a name plus sequence, and it should print a 1/XX, 2/XX, etc. on the images (make sure Photo Info is checked and you choose "Sequence" as an option). If you choose this option, be sure and select where you want the images (I would suggest Desktop) and once you click Print/Print LR will create a folder with the images all named and in sequence (in the file names) and also have the 1/xx, 2/xx, on the image file. You may have to play with the font color for the name printed on the image file, since it defaults to black. Just not sure if that is what you want vs. a unique filename for each photo.

So exactly what, Mr. OP, are you looking for?

You say "as shot," which may mean that the file names are already in sequence and they get scrambled when uploaded to Smugmug?  If so, just go to Arrange/Sort and sort on the file names.  You can also sort on time taken.

Alternatively, and simpler, in Smugmug at the top of the gallery you have "1 of X" as in "1 of 31" images in this gallery, so just use that gallery number to talk about the right images.

Hey ... I wasn't causing trouble ... just pointing that the OP mentioned they wanted to ... "Is there a way I can print sequential serial numbers/ids on my images" ... and I offered options to do so ... sorry to intrude ...

Another approach is to use Slideshow module. Here is a blog post from John Nack from Adobe: Lightroom: Watermark your images with sequential names/numbers

Or use Visual Watermark app: How To Sequentially Number Images

The OP was pretty clear that he wanted numbers on the actual image. Just because you read it wrong and were corrected doesn't mean you need to throw a hissy fit.

You're three years late to the party, David...