I don't have statistics handy, so I don't know what percentage of landlords want a paper check. But it's certainly not unheard of. Especially if the landlord is a small time operation with just one or two properties.
As to can you get checks without going to a physical bank, sure. I haven't gone to a bank office to buy checks in decades. There are companies that sell checks on-line. As part of the order process you have to tell them your routing number and account number so they can print it on the checks. But you order on line and they mail them to you. My bank's web site has a menu option to order checks on line. I've never used it so I don't know the process, but I don't have to physically go to the bank even if I want to order from the bank.
Another option you may have is to have the bank mail them a check. My bank lets me direct them to make payments on their web site. I get on the web site and say to send however much money to this person or company, this address, and here's my account number. If the person or company I send it to has things set up with the bank, they'll do an electronic transaction. Otherwise they print and mail a paper check. I can see on the web site which way they sent the money, but it doesn't matter. I don't do anything different. And if some business I'm paying signs up to get the money electronically, they'll just automatically switch from sending paper checks to doing it electronically. I don't have to know or care.
I make contributions to my church and my cleaning lady this way all the time, so I don't have to bother hand-writing a check, putting it in the mail, and paying postage. I've used this to send money to relatives. Etc. With my bank, it's a free service. They don't even charge me for the postage to mail the check. Probably because they save more by not having to pay someone to handle the check then the cost of printing and postage. I wouldn't be surprised if some banks charge some sort of service fee for this.
These days I almost never actually write a paper check. The only people I can think of that I send a paper check regularly is my local newspaper: they don't have any way to pay online, and I can't have the bank send them a check because there's a form where I have to specify some details. Everybody else I either pay through the recipient's web site or I pay through the bank's web site. At one point I moved and I didn't even bother to order checks with my new address on them for 8 years, because I was writing so few checks it was easier to just cross out the address and hand-write the new address. :-)