Did the Honorable East India Company (HEIC) operate in the West Indies as well as the East Indies and, if so, did they face competition from other English or British chartered companies?
This is a tricky question to answer definitively given the complexity of the 250+ years of history of the East India Company in its various forms (referred to here, for simplicity, as the HEIC). While I think it would be impossible to prove that no single HEIC ship ever visited any part of the West Indies, I think it is safe to say that they didn't 'operate' in the West Indies (i.e. they didn't trade regularly or have any permanent presence there).
The English (and later British) Joint Stock companies that were created for foreign ventures split into two types. The 'old world' ventures, like the East India Company and the Levant Company, were essentially trading companies where trading posts and colonies were created as a by-product of establishing trade. The 'new world' ventures, such as the Virginia Company, existed to create colonies (or "plantations") where trade was a by-product of the colonisation. So there was a difference in ethos between the companies trading to the East and West.
The HEIC spent most of its existence fighting to protect its trade to the East Indies. While, at times, it had the theoretical advantage of possessing a monopoly on the trade, this monopoly only existed in terms of English (and later British) competition. The Portuguese, French and Dutch (especially) had their own East India companies each of which was competing to bring trade back to Europe and prevent their competitors from doing the same. Not only that but it had to fight constant political battles to protect this priviliged position in Parliament and in the Royal court. While the East Indies trade made the HEIC considerable amounts of money, much of this was reinvested in protecting its position from all comers (to the point that it was on the verge of bankruptcy on several occasions).
Had the HEIC tried to branch out into trade with the new world, it wouldn't have had the advantage of a monopoly (which allowed for significantly higher freight rates) so the trade it had would have been significantly less profitable than trade to the East Indies. Since it always had a limited number of ships available, therefore, it would make little commercial sense to use their time on significantly less profitable trips to the West Indies. Also attempting to spread its influence to other parts of the world would probably have provoked calls from other British trading companies to remove the monopoly the HEIC had to the East.
Lords of the East: The East India Company and its Ships, Jean Sutton (Conway 1981)
The Trade of the East India Company, F. P. Robinson, (Cambridge UP, 1912)