The semi-canon answer is that he joined of his own free will, but under a particular delusion about their arrangement.
The Shadow Within is a B5 novel that covers (alongside other plots) events on Z'ha'dum and the Icarus before Babylon 5 came to be. The brief summation of Morden's backstory is that he was a linguistics doctorate with some shady connections to Earthforce who had recently lost his family. Their ship had been passing through the Io jumpgate when the jumpgate was bombed and the vortex collapsed in on the ship. Without their bodies for closure, he became unable to mourn and move on. After becoming completely unhinged at the news, he adopted the constantly smiling attitude we see in the TV series, as a way to control his continuing emotional breakdown. Anna remarks during a meeting:
Morden's smile was sometimes genuine, sometimes not. The trick was figuring out which and when.
He had half-convinced himself, caught between denial and despair of their death, that his wife and child were still alive somehow, caught in hyperspace.
"Of course I should have been with them. They found debris, but not enough to account for the whole ship. They said some of it must have been drawn through into hyperspace. They said no one could have survived. I know they're right. Half of a blown-up ship in hyperspace. But sometimes I wonder if they could be alive. And then I wonder what it could be like, floating through hyperspace, lost, alone. Sarah would be six now."
Initially Morden is a kind-of ally of Anna as the Icarus expedition devolves into a nest of intrigue (Interplanetary Expeditions, Psi Corps, and Earthforce through Morden, all have their own interests and plots) When everything is finally going to hell and Anna and Morden try to make a final stand against being taken, the Shadows seize on Morden's inner emptiness and send him a vision of his wife and child falling through a crushing void, stuck in a pocket that loops in on itself endlessly. Morden caves under the anguish of the vision and pledged loyalty if only the Shadows would release them from this fate. The next vision he gets is of a Shadow ship destroying the pocket with a laser beam, his family's pain ended. Given Morden's mental state and later morals, he appears to have taken these visions as the confirmation and closure he couldn't get elsewhere, and while the emptiness and constant controlled smile has remained, Morden has continued to serve the Shadows willingly ever since.
There are only some small reminders of this plot elsewhere. The pendant he carries is a love token stone from a lost civilization that he bonded with his wife over, that Anna worked into a pendant for him in an attempt to draw him out of his shell. That's about the only connection made between this plot and the TV series.
Both Elric and his apprentice Galen investigate Morden's motives over the course of The Passing of the Techno-Mages trilogy. News footage is found of Morden's breakdown, and there's discussion of the motive behind the bombing (the perpetrators were never found, suggesting Morden has a revenge wish) but nothing ever comes of it. Elric also briefly steals Morden's love token, to get a reaction, so it's pretty clear Morden's motivations are still the same, that he hasn't been mind-wiped or altered beyond what might be considered normal.
The furthest Elric and Galen get to penetrating Morden's mysteries is when Elric detects a brain implant on Morden, linking it to something the Drakh also have, believing it allows him to receive communications directly from the Shadows. Galen later comes to believe that Morden's emotions are kept in check or manipulated by the same Shadow implant, acting like a pleasure center most of the time when he's on task, keeping his manner smooth and his smile stable. He attempts to engage Morden by killing his handlers, interrupting the implant signal, and offering him freedom or death. Morden is rattled and admits to being manipulated, both by his implant and by the memory of his family, but insists what the Shadows gave him was worth the price and refuses to switch sides. More Shadows converge moments later, and Galen flees without finishing the job.
So the upshot of all of the semi-canon novels is that Morden is with the Shadows of his own free will, but the Shadows keep an eye and a lever on him. You might think of his tag alongs as a combination of handlers, controllers (if necessary), observers, and bodyguards.