Is there any risk of illness from putting ketchup or mustard on my sandwich?

Probably not. Though ketchup is made from tomatoes, a potential sources of E. coli and salmonella, the tomatoes are cooked during processing, thereby eliminating the two pathogens. The same goes for spices, also potential sources of E. coli and salmonella.
 
Likewise, spices, powders, and other ingredients contained in mustard are potential sources of E. coli and salmonella. Once again, cooking kills the two pathogens during processing.
 
Because suppliers of spices and the like irradiate product before shipping them, it is unlikely these ingredients would arrive contaminated at a ketchup or mustard manufacturing plant.
 
Because of their natural acidity, ketchup and mustard are highly resistant to spoilage, one of the reasons both store so well at room temperature. Both also contain vinegar, which helps to prevent spoilage.
 
For purposes of quality, manufacturers recommend refrigerating ketchup and mustard once jars have been opened.

Source(s):
Food Safety Central "Ketchup, Anyone?"
Myheinz.com "Frequently Asked Questions"