"the E-sail relies on charged particles such as protons and alpha particles in the solar wind. The idea is to generate an electric field around the spacecraft which deflects these ionised particles and generates a force that accelerates the craft throughout its journey."

I'm curious if this type of vehicle can travel toward the sun? My initial impression seems to be that it's forever destined to be repeled by the sun.

Not repelled by the sun, but propelled by the prevailing local source of charged particles as long as the sail is deployed.

Hypothetically, if that source is on the opposite side of the vehicle to the sun, then it could be propelled towards the sun until such point as the sun became the prevailing source, at which point, the sails could be stowed and the vehicle could coast until such time as it needs to decelerate. Then it could deploy the sails to slow down.

Certainly, travelling towards a major source of charged particles will take longer than going towards a weak one, as you would only be able to accelerate as long as your nearer source is stronger.

...which was described in Science Fiction as early as 1951. Probably the first in a story was one by Cordwainer Smith. He wrote some really innovative stuff, I recommend a read.