He would've sent an owl from Diagon Alley.
Muggle-borns generally get a visit from a Hogwarts member of staff. Riddle got Dumbledore. Harry got Hagrid. The communication issue can probably be put down to a blunder on Hagrid's part. He probably should have just knocked on the door and asked to speak to Harry and explained the situation in person. However, he may have had some recollection of McGonagall's opinion of the Dursleys. Knowing that they are particularly Muggleish Muggles he may have decided that simply popping the letter through the letterbox was the wisest approach. When that failed, for reasons best known to himself, he decided to inundate the Dursleys with increasing quantities of letters.
As for how Muggle-borns traditionally signify their acceptance of an offer, they have to send an owl. This can be done from Diagon Alley. Of course, if a teacher presents their invitation in person then they can presumably accept there and then. The owl is just a written confirmation. The teacher would explain that the child has to go to Diagon Alley (with their parents, if necessary) to buy their school supplies.
“Where do you buy spellbooks?” interrupted Riddle, who had taken the heavy money bag without thanking Dumbledore, and was now examining a fat gold Galleon.
“In Diagon Alley,” said Dumbledore. “I have your list of books and school equipment with me. I can help you find everything —”
(Half-Blood Prince, Chapter 13, The Secret Riddle).
Even if the teacher doesn't offer to go with the student, as Dumbledore did, the Hogwarts letter itself contains the full list of required items. The teacher only needs to give the family details of how to get to Diagon Alley.
“Still got yer letter, Harry?” he asked as he counted stitches.
Harry took the parchment envelope out of his pocket.
“Good,” said Hagrid. “There’s a list there of everything yeh need.”
Harry unfolded a second piece of paper he hadn’t noticed the night before, and read...
(Philosopher's Stone, Chapter 5, Diagon Alley).
If the Dursleys were supportive of Harry's decision to go to Hogwarts (as the question suggests) then they presumably would've had no problem with taking him to Diagon Alley.
Once in Diagon Alley you can buy your own owl.
Or, if you don't want an owl, you can hire one from the post office. Diagon Alley presumably has an owl post office; Hogsmeade, which is rather smaller, has one after all.
“Thanks,” said Harry, picking up a packet of tiny black Pepper Imps. “What’s Hogsmeade like? Where did you go?”
“The post office, Harry! About two hundred owls, all sitting on shelves, all color-coded depending on how fast you want your letter to get there!”
(Prisoner of Azkaban, Chapter 8, The Flight of the Fat Lady).