"Vulnerable" already covers the RAW flammability of a golem. Even if we choose to treat their body components as objects rather than a creature, they are certainly both "wearing" and "carrying" their own bodies. That ignition exclusion to worn an carried objects is obviously in place to prevent fire from breaking other game mechanics (making too many combats trivial, and frustrating players by too-easily depriving them of gear). But it also makes intuitive sense as an active creature can reasonably easily extinguish a flame that's just started and isn't being fed by an accelerant such as Alchemist Fire.
All of this supports the consensus of other answers that a wooden golem would not ignite at all. I'd take it further and posit that large masses of wood are not, in fact, particularly flammable. If I apply a blowtorch to a wooden bench I'll do much more damage to its substance than if it were metal or stone, which is reflected in the golem taking double damage from fire. But when I take the blowtorch away, the bench will not be ablaze. It takes a good deal of heat over a sustained time to induce a decently thick board or branch to truly and sustainably ignite. This is why bonfires have to be carefully constructed with tinder and kindling and a flow path for oxygen in and smoke out. Houses only burn because they're constructed in a way that coincidentally provides all those things.
So unless your wood golem is woven out of wicker and thin dry twigs, it won't ignite-- in game or out. I might have the lacquer on a jointed Pinocchio-style mannequin ignite mostly for visual flavor, but after a round or two when the lacquer had burned away my golem would look grotesque but be largely unharmed in any HP sense. An animated angry log would only take its double damage and some scorch marks.