If you make friends with the sawmill owner (by doing a quest for them selling them fire wood etc) some will let you saw your own logs and the logs will be sent to you homestead. There are some bugs though copy/paste from the wiki page -
Sometimes when asking to cut sawn logs at a mill when in the mill owner's house, they will say that they can't give it away.
Logs may appear in the inventory under Misc with the appearance of Elven Boots.
Upon cutting logs at a mill, no logs are awarded. Asking to buy lumber and choosing, "Can I cut my own?" will grant lumber again. (Tested with Hod at Anga's mill after several attempts with no lumber.)
Oh, I'm glad others have had the same problem as I have. I'm at Half-Moon Mill and kept talking to Hert and she kept telling me it was okay to cut the lumber and that she needed the break or it was okay and it'd even be delivered for free. It took me at least 8 tries before I finally started getting prompted that 10 logs were being added.
Once it started working, every cut since I've been doing it has worked successfully. Annoying bug to say the least, so I've been cutting logs repeatedly to make sure I have enough for the rest of the house. While wating, I've been building Alteration through Candlelight repeatedly to not completely waste time. ;)
Just going to add I'm on 360 and that Hert has been mining nearby the whole time; whenver she tries to leave, she turns around and goes back to mining.
Don't know if it's worth mentioning, but Hert did walk away from mining briefly towards me and I didn't get credit for that log I was working on. She then turned around and went back to mining and the logs started adding again. I don't know if the NPC has to be preoccupied with something else (at least in Hert's case).