While more can always be written, you may be interested in this previous response of mine to a similar question, which links back to an even earlier response of mine to another related question.

The TL;DR answer is that the colonists were never looking for seats in Parliament. They had representation already, in their local colonial legislatures. Their colonial charters (something analogous to state constitutions) in many of the colonies stipulated taxation rights lay with the local legislature. This was the case with Massachusetts, where the taxation issue first broke out. A more accurate slogan would have been "No taxation outside our local representation".

One source I neglected to mention in those earlier responses is the book The Ideological Origins of the American Revolution by Bernard Bailyn. In that book, the author talks a bit about the argument on both sides. On the British/Tory side, the argument was in favor of "virtual representation", that not even in Great Britain were most British people represented, except in a "virtual" sense, that all MPs were supposed to represent the people of the whole country, not just their constituency.

As Bailyn points out, there was criticism by Americans of this system even as it operated in Britain. Bailyn states that Parliament actually did begin as a body that was supposed to be of semi-representative democracy, and that each MP was supposed to represent the interests of only their particular constituency. The idea of "virtual representation" only evolved over a long period of time. But at the same time, the concept of representation in the American colonies evolved in the opposite direction. The representatives in the colonial legislatures were often elected by a local constiutency of affluent landholders, but who outlined an agenda through town councils and such which the representative was supposed to represent. In other words, the representatives were very much sent to the General Assembly (or whatever its name may be) to represent local interests, not continental American interests or British interests.

Thus, when the arguments leading to the American Revolution broke out, American political thinkers found it easy to criticize the concept of "virtual representation".

The only representation the Americans were ever really interested was their local representation, as it had existed since the 17th century. In other words, they wanted home rule, on many tax issues as well as on other issues. My earlier answers address this part of the question. While there were some proposals to get the Americans some representation in Parliament and/or at the British national level, Americans were generally against any plan that would weaken their argument for representation and power in their colonial legislatures. The more generous plans, like ones giving the American colonies a representative in Parliament who would have veto power over tax legislation as it applied to the American colonies, were generally proposals that the British government and crown would never agree to, either.

This position of the British government only changed after the war had already begun. In October 1778, the Carlisle Peace Commission offered a home rule scheme and representatives in Parliament of the American colonies to end the war. But this was rejected, because by then, there was a real sense that the U.S. could win the war. France had already entered the war on the Americans' side and Spain was offering assistance, too. On top of that, there was never any guarantee that Parliament couldn't rescind some of the home rule or representation at some later date, when the Americans and the crown found themselves in another political firestorm. The Carlisle Peace Commission proposals were seen as too little, too late. The Americans continued fighting the war.

EDIT:

One last thing to point out is that "no taxation without representation" was only one of the grievances that the Americans were upset about that led to the war, and its major flashpoints were actually years earlier. Read the whole Declaration of Independence if you want to read the other grievances. Some were more important than others, but it was really the "Intolerable Acts" that got the states to unite for the Revolutionary cause. Those acts unilaterally dissolved the government of Massachusetts, shut down the port of Boston effectively shutting down their economy, and put them under martial law until the colony agreed to pay some back-taxes that Parliament believed the colony owed. British soldiers patrolled Boston and could not be prosecuted locally for any actions they took. If accused of doing something illegal, they would be sent back to England to face any charges - or no charges at all. This went above and beyond what any of the colonies believed was just, and the implicit threat in the acts was that the same could happen to any of the other colonies if they crossed the British government as well.

It was then that calls of "tyranny" and openly questioning just what kind of government the colonies were living under really gained popular acceptance in the minds of a majority of Americans. On something like taxes alone, Americans could still be convinced. But on something like dissolving the government and martial law with British soldiers patrolling the streets and not even being held accountable for any of their actions, that was what was considered "intolerable" to most Americans.