By Derrick E. Jean-Baptiste
Yes. I'd go as far as to say that not only did he have a legitimate shot, there were certain points where I'd honestly say he had the best odds to take the throne.
It's important to remember that Littlefinger was playing the long, looooong game. He had been planting seeds since before the show even started to gain more and more power, and it took literal magic to figure the full extent of these plans out.
Look at Littlefinger's position in Season 7. He controls the Vale, the only one of the Seven Kingdoms virtually untouched by war in the last several years. He is also Lord of Harrenhal, which, while a ruin, held significant lands in the Riverlands, and after the destruction of House Frey, he was one of the most powerful lords in the Riverlands.
Jon left Sansa in control of the North, and the Northern lords were clearly starting to lean more towards Sansa. It would have been easy for Littlefinger to say to the lords, "Jon Snow, your king, has kneeled to the foreigner daughter of the Mad King" and that would have been it for Jon. Littlefinger's plan was to get Sansa in control of the North, marry her, and get another kingdom under his belt. Remember, the only reason this plan failed was because of literal time travel magic that revealed just how much he betrayed the Starks over the years.
So Littlefinger marries Sansa, and right there he has like 2.5 kingdoms. If he swooped down into the Riverlands and say, rescued Edmure Tully with his niece by his side, the rest of the riverlords surely would have joined him.
So then the factions would have been:
1. Dorne, the Reach, and half the Iron Islands. Led by the daughter of the Mad King who hasn't even set foot in Westeros her entire life and brought three dragons, an army of slaves, and a Dothraki horde to the realm.
2. The Crownlands, the Westerlands, some of the Stormlands, and half the Iron Islands. Led by the incestuous queen who did the Westerosi equivalent of 9/11 and made it a personal goal of hers to get her subjects to fear and hate her as much as possible.
3. The North, the Vale, and the Riverlands. Led by a native Westerosi. The guy who faithfully served on good King Robert's small council. The guy who married good old Jon Arryn's widow and is the stepfather to his heir. The guy who married the honorable Ned Stark's daughter.
It's hard to not give him at least a fighting chance in that scenario.