Can mosquitoes or other insects transmit Lyme disease?

As far as we know the tick is the only vector. The Borrelia bacteria has a complex relationship with both the tick, in which it spends part of its life, and the mammalian host (mouse, bird, human etc). Although the Lyme disease bacteria have been found in the salivary glands of mosquitoes, there is no evidence that mosquitoes are capable of passing them on in a very short bite, in the number and state required for an infection. When a tick attaches to a mammal, it takes some hours before the Borrelia are passed, and the bacteria change their outer membrane in the process.

It is possible there are other vectors but experiments so far have failed to achieve transmission.