Does rearrangement take place in the reaction between ethenylcyclohexane and iodine chloride?
Yes, I believe the rearrangement can happen. However, the product from the rearrangement is probably not the major one.
The closest reaction to what you have drawn that I could find on SciFinder is this:

The paper is mentioned below in the reference. In the paper, the researchers performed similar reactions of alkenes with $\ce{I-Cl}$ and in cases where there is a possibility of a rearrangement to a tertiary carbocation, the by-product similar to $\text{(C)}$ is generated (for example in case of the cyclohexane ring being replaced with iso-propyl, the percentages are $\text{(A)=55%; (B)=2%; (C)=38%}$).
For the reactions drawn above, the researchers did not mention the A:B ratio, however, in most cases A is the major product, which is expected (Markovnikov's rule). Nevertheless, the product formed from the rearrangement is not the major one in any of the reactions.
The mechanism is not mentioned in the paper, but my guess is that it will not be much different from the usual carbocation rearrangement mechanism. Only in this case, the hydride shift will break the iodonium ring, while creating a tertiary carbocation. The Cl can then attack the carbocation and form the product.
This is my proposed mechanism:

To reiterate, this is likely not the major pathway. The usual addition to the double bond will be the major pathway and form the major product.
Reference:
T. Horibe, Y. Tsuji, and K. Ishihara, &quot;Thiourea–I2 as Lewis Base–Lewis Acid Cooperative Catalysts for Iodochlorination of Alkene with In Situ-Generated I–Cl&quot;, ACS Catalysis, 2018, 8(7), 6362–6366