The problem is not really about products, more about finding an arrangement of digits 1 to N-1 (for base N, since digit N does not exist in base N), that conforms to the criterion that any pair in the arrangement is part of the set of single digit products.
There could be a better way to solve this, but I wrote a small program that creates a table of possible digit combinations and uses that to try to find a valid arrangement.
It's not optimal in terms of performance (essentially a depth-first-search), but finding a pattern for base-N takes N times longer than for base-N-1.
this My program was able to show that bases 3, 4, 5, and 6 do not have such an arrangement (due to digits that do not exist in a product).
Interestingly, for base-2 (binary) it thinks that "1" is a valid combination due to the only non-zero digit being 1, so the row has no neighbouring digits.
Otherwise it shows that bases 7 to 21 have possible arrangements of digits.
base-2: 1 (questionable)
Here's a link to a program you can run in your browser.