I can't speak to how common or widespread they were. Nor if there was anything resembling what we would consider a modern day company. But, there were certainly places in the past were you could rent horses. Here is a passage from the newspaper "The Spectator" from 1712:

Mr. Tobias Hobson, from whom we have the expression, was a very honourable man, for I shall ever call the man so who gets an estate honestly. Mr. Tobias Hobson was a carrier; and, being a man of great abilities and invention, and one that saw where there might good profit arise, though the duller men overlooked it, this ingenious man was the first in this island who let out hackney-horses. He lived in Cambridge; and, observing that the scholars rid hard, his manner was to keep a large stable of horses, with boots, bridles and whips, to furnish the gentlemen at once, without going from college to college to borrow, as they have done since the death of this worthy man.

I say, Mr. Hobson kept a stable of forty good cattle, always ready and fit for travelling; but, when a man came for a horse he was led into the stable, where there was great choice, but he obliged him to take the horse which stood next to the stable-door; so that every customer was alike well served according to his chance, and every horse ridden with the same justice: from whence it became a proverb, when what ought to be your election was forced upon you, to say Hobson's choice. This memorable man stands drawn in fresco at an inn he used in Bishopsgate-street, with an hundred pound bag under his arm, with this inscription upon the said bag:*

"The fruitful mother of an hundred more."

Whatever tradesman will try the experiment, and begin the day after you publish this my discourse to treat his customers all alike, and all reasonably and honestly, I will ensure him the same success.

-The Spectator, “No. 509.” Oct. 14, 1712. From Project Guttenberg. http://www.gutenberg.org/files/12030/12030-h/12030-h/SV3/Spectator3.html#section509 (Accessed 8/27/2016).

This passage was written about 80 years after Tobias Hobson lived, but I believe what it does show is the fact that the concept of renting horses was known. The reason Hobson was of note was his practice of rotating the horses to keep them from being overworked. This is incidentally the origin of the phrase Hobson's Choice, which means one is given the option of either accepting what is offered or refusing the offer. Essentially "take it or leave it".

Yes, it was very common!

In 18th and 19th century England and America, most stables hired horses on a regular basis. Depending on how wealthy the renter was and how long the horse was needed for, the "rental" could look very different.

1. Renting livery. This meant hiring a horse or small team and a cart, carriage, or sleigh. Smaller, nicer stables catered to the rising merchant class and falling nobility: anyone who needed to look spiffy but couldn't afford to stable their own horse and vehicle. A moderately successful businessman or member of the minor gentry could rent "livery" from a stable. This meant a horse and carriage, sometimes with servants in the rentee's livery.

For example, the R.G. Dun Credit Reports at Harvard's Baker Business Library records "livery rental" become code for "horse and equipment rental" in Boston. The spring "courting season" saw a sharp increase in livery rental. It's similar renting a limo to take your date to prom.

2. Hiring horses for labor. This was like renting a U-Haul or Home Depot truck. In large, public stables, anyone could hire a horse for a day. Poorer farmers could hire a horse for spring plowing; slightly more well-to-do farmers could hire a team of horses or oxen. Some London cabmen also hired their horses, either on weekly, monthly, or pay-to-own plans.

The Dun reports record poor conditions and overwork in large public stables. They're corroborated by protest writing in England like Anna Sewell's 1877 protest novel Black Beauty. Based on Horace Bushnell's "Essay on Animals," it documents various cruelties inflicted on horses, including hired cab horses.

3. Hiring post-horses. Post-horses belong to mail posts and were used to deliver messages and the mail. They could be hired for short, fast trips, usually between posthouses. Since most public stagecoaches stopped at posthouses, it was easy to hire a post-horse. (For example, if you arrived at the posthouse but needed to go 10 miles out, you could hire a horse and boy to ride it back).

Marcus D. Hanover's 1875 "A Practical Treatise on the Law of Horses" has a long chapter on ethical hiring practices. Without prior agreement, hirers are not entitled to choose a particular horse. A man who hires a horse and tries to resell it will be tried under felony law for fraud and attempted theft. If the horse is trained as a racehorse, the owner does not have "a right to take the animal away for the purpose of running races for plates elsewhere." Etc.

Sources: Clane McShane and Joel Tarr. The Horse in the City: Living Machines in the Nineteenth Century. John Hopkins University Press, 2007.

They were around especially in larger towns/cities where the average person wouldn't need a horse on a regular basis so it was more economical to rent one occasionally (much like in some large cities today people can get away without owning a car). Unfortunately, I can't speak to any statistical evidence in terms of how many were in various states.

As far as a specific example of renting a horse, John Wilkes Booth and David Herold rented a couple horses for their escape from DC after Lincoln's assassination. Those horses became stolen though after the two skedaddled from the capital. The horses were only rented for a few hours in this case. The livry owner actually saw Herold (I believe) at one point and yelled to him that the horses were supposed to be back very soon (this was just prior to the assassination).
Some more on this particular event is in James Swanson's Manhunt. Great book. :)