Kyoto Imperial Palace logo

Why are the products sold in the Kyoto Imperial Palace (for example, traditional Japanese bookmark, but I am not sure if the bookmark is really an example) uses a carriage as design and not the chrysanthemum?

It's interesting...
An ancient carriage like this?
http://tabelog.com/kyoto/A2602/A260202/26016212/dtlrvwlst/4634119/1557...
It's 牛車(Gissha; ox carriage) and expressing the ancient Imperial court, because samurais ride a horse in the later eras.
To begin with, the shops and souvenirs have no connection with the Imperial Household, so it's simply selected according as design, I guess.

But there are some wagashi that are shaped like chrysantemum and seem to be sold by Imperial Household staff

while the traditional Japanese bookmarks and other products have carriage as design

Sorry, "no connection" is overstatement.
Of course, temples/shrines and shops that use 菊の御紋(kiku-no-gomon; the Imperial chrysanthemum emblem) have historical connections with the Imperial Household and they were permitted to use the emblem before.
The shops in the palace are managed on the commission for the government office, because today after WWII, the Imperial Palaces are nationally-owned not the Imperial estate.
Though that's not reason why it's not chrysanthemum but carriage...
If in Tokyo, I can ask them directly.(x_x;)I'm interested in too.