Sure, it would have been possible to write to different establishments telling of your impending arrival, but that's not how you would have gone about it.

The short answer is that you would have made arrangements upon your arrival, inquiring with the locals to find a suitable place to stay. You may have gone to a tavern to do so (which often doubled as inns), or maybe you just asked someone on the street.

Take the example of someone like Benjamin Franklin. He made several voyages to England and France during the 1700s. The first occurred when he was 18 years old, when he went to London in order to buy some printing equipment for his business. He didn't have much money, and as it turned out, the governor of Pennsylvania (under whose guidance he had made the trip) hadn't followed through on a promise to make credit arrangements, and Franklin found himself stranded for about 18 months until he earned enough money to pay for a return voyage to Philadelphia.

So, at the start, Franklin and his friend James Ralph simply found a cheap place to stay, with no prearrangement:

[James Ralph and I] took lodgings together in Little Britain at three shillings and sixpence a week; as much as we could then afford. He found some relations, but they were poor, and unable to assist him.

Source: The Autobiography of Benjamin Franklin, page 58

How exactly they found their lodgings is not written about, but again, it would have been customary at the time to inquire at a local tavern.

When Franklin returned to England in 1757, he went as a representative of the Pennsylvania Assembly. Shortly after his arrival, he stayed with a widow named Margaret Stevenson on Craven Street in London, and spent most of the next fifteen years there. Her house "had been recommended to him by some Pennsylvania friends, who had lodged there." (Source.)

Where exactly he stayed the first night he got off the ship is unknown, but as a diplomat, he would have written ahead, telling of his impending arrival, and would likely have had some counterpart in England meet him at the dock. His English counterpart would have made some arrangements for his initial stay before he had paid his visit to Mrs. Stevenson.

Maybe more informative is exactly how within their own country people like Benjamin Franklin and John Adams made arrangements for lodgings when staying the night away from home.

In 1723, at the age of 17, Franklin relocated from Boston to Philadelphia. In his autobiography, he writes of his arrival in Philadelphia, and found his first lodging there by asking someone on the street:

I was dirty from my journey; my pockets were stuffed out with shirts and stockings, and I knew no soul, nor where to look for lodging...Walking down again toward the river, and looking in the faces of people, I met a young Quaker man, whose countenance I liked, and, accosting him, requested he would tell me where a stranger could get lodging. We were then near the sign of the Three Mariners. " Here," says he, "is one place that entertains strangers, but it is not a reputable house; if thee wilt walk with me I'll show thee a better." He brought me to the Crooked Billet, in Water Street. Here I got a dinner, and while I was eating it several sly questions were asked me, as it seemed to be suspected from my youth and appearance that I might be some runaway.

After dinner my sleepiness returned; and, being shown to a bed, I lay down without undressing and slept till six in the evening, was called to supper, went to bed again very early, and slept soundly till next morning.

Source: [The Autobiography of Benjamin Franklin, pp. 39-41] (https://archive.org/stream/autobiographyofb02fran#page/38/mode/2up)

During the Revolutionary War, John Adams was often travelling throughout the country, staying the night far from home. As is evident in his letters to his wife, he wasn't planning sleeping arrangements beforehand, but was making them from town to town as he arrived there.

One such letter on a stay in Fishkill, New York, in 1777 starts out:

After a march like that of Hannibal over the Alps, we arrived last night at this place [Fishkill], where we found the utmost difficulty to get forage for our horses, and lodgings for ourselves, and at last were indebted to the hospitality of a private gentleman, Colonel Brinkhoff [Brinckerhoff], who very kindly cared for us.

(Source.)

Keep in mind that John Adams was about as high profile a person you could meet in America at that time. He was a member of the Continental Congress, yet he was arriving in towns and making sleeping arrangements on the fly.

TL;DR: No, you wouldn't book long term arrangements beforehand, nor would you send letters to individual establishments. If you were a diplomat or some higher up, you might have someone at your arrival point meet you, to whom you would have written beforehand. They would have taken care of your sleeping arrangements for the first few nights. From there, you'd make arrangements on your own.

If you were working class, you'd inquire about sleeping arrangements upon your arrival.