Reaction mechanism for 7-hydroxy-4-methylcoumarin synthesis with an iodine catalyst
There is a paper about this exact reaction, but in addition to the use of $\ce{I2}$ it mentions the use of microwaves. In the reaction, $\ce{I2}$ acts as a Lewis acid, catalyzing both the transesterificaiton and electrophilic aromatic substitution reactions. It can be seen acting as such in the first step of the mechanism:

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Next, a proton transfer occurs:



Next, the resorcinol derivative can act as a nucleophile, and ethanol is ejected, completing the transesterification reaction:





Next, $\ce{I2}$ coordinates to the keto carbonyl, activating the carbon towards electrophilic addition. Because this resorcinol derivative is so activated by the $\ce{-OH}$ and $\ce{-OOCR}$ groups, a weak Lewis acid like $\ce{I2}$ is sufficient to catalyze the reaction:



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