You do not need a car to explore the Dales although it remains the easiest way to get around this vast area.
There are many bus routes in the Dales and by using a rail head such as Harrogate, Northallerton or Skipton it should be possible to use the buses to tour the Dales.
Hawes can be reached by bus and the link to the official website has a lot of information. Please contact me if you need specialist advice or more links.
For many years, Wensleydale's 19th century railway line to Hawes from Northallerton and Leyburn has been closed but thanks to some hard working locals, it looks brighter than in a long time as now the trains have at least started running from Leeming Bar in the Vale of York to Redmire which is just at the lower end of Wensleydale. This so far doesn't help if you are going to Hawes but hopefully, they will succeed one day and have formed an association to lobby more actively. At least it would help if they could manage trains to and from Northallerton to Leyburn to tie Wensleydale to the rest of Britain's network. Who knows. Keep an eye on the website below for the summer timetables and what's new.
It's not easy but it works. There are buses from Leyburn (reached from Northallerton along the railway network, or from Richmond or Ripon) through Wensleydale which all stop here. Some of them are only Royal Mail post buses and can get crowded so you need to look in the timetable but if you go for Dales & District departures, you can buy a ticket allowing you a full day's travel. All buses stop in the Market Place (just note that there is only a bus stop on one side but they stop on both) and the post bus also stops outside the post office in this picture but there is easy walking distance between the two should you be in a hurry and confuse places.
Summertime, there is sometimes a veteran bus between Hawes and Ribblehead at lunch. Ribbelhead has its famous viaduct along the Settle to Carlisle railway which you can see, as well as the station museum. Apart from that, you get very nice Dales scenery and there is of course the added bonus of the vehicle if you like that sort of thing.