Yes, they can—especially mosquitoes. Mosquitoes do not need a pond; they can breed in any small amount of stagnant water. If an urban planter box drains poorly, water can collect at the bottom, in a clogged drainage hole, in the saucer, or in constantly saturated soil. Female mosquitoes may lay eggs there, and the larvae can develop quickly in warm weather.
Poor drainage can also make a planter box more attractive to other pests. Fungus gnats thrive in wet potting mix, drain flies are drawn to stagnant water with organic matter, and ants or cockroaches may shelter in damp gaps. The box itself is not the problem; the standing water and consistently wet conditions are.
To reduce the risk, keep drainage holes clear, empty saucers after rain, use a well-draining potting mix, avoid overwatering, and check the box after storms. If standing water cannot be removed, use mosquito dunks or other BTI products according to the label. Good drainage and regular checking are the simplest ways to keep urban planter boxes from becoming mosquito breeding sites.