This page provides a list of online shops that sell tools and materials for Japanese woodblock printmaking / mokuhanga.
A couple of points first:

• This is not an endorsement of the linked websites. Shop at your own risk.
• I am not affiliated with any of the linked websites.
• This is mostly geared towards US-based shoppers; but there are some European and Asian sites as well.
• I will try to tag each merchant with the types of products that they sell, so you can just search for [paper] or [chisel] for example.
• This is not an exhaustive list. If you know of a good site for mokuhanga products that’s missing from this page, let me know in the comments. Maybe I’ll add it.

Without further ado, here is the list:

McClain’s [US]
McClain’s is a great starting point for supplies. They sell pretty much everything you’ll need… Paper, blocks, carving tools, etc.
[baren][blocks][wood][brush][knife][chisel][ink][paint][paper]

Washi Arts [US]
Washi Arts is primarily a retailer of Japanese paper, they also sell some printmaking tools.
[paper][blocks][wood][knife][chisel][baren]

Hiromi Paper [US]
Hiromi Paper carries a big selection of Japanese Papers, many of which seem suitable for printmaking. They generally have decent product photos and descriptions. Their downloadable PDF catalog is particularly useful — with info about the paper makers, information about each paper, a glossary of terms, etc.
[paper]

Hida Tool [US]
Hida Tool focuses primarily on Japanese hand tools, but they also carry Michihamono brand chisels, brushes, and barens; as well as Power Grip brand carving tools.
[knife][chisel][baren]

Tanuki Prints [US]
Tanuki Prints sells barens made by John Amoss.
[baren]

Dick Blick Art Materials [US]
Blick has a huge selection of general art supplies; including some papers that are suitable for various printmaking techniques. Of course, they sell paint, brushes, and various other accessories that might be useful for printmaking.
[paper][paint]

Ocooch Hardwoods [US]
Green Valley Wood Products (also their Ebay Shop) [US]
Cherokee Wood Products [US]
Woodchucks Wood (also their Amazon Store) [US]
Wood Hawk Lumber (Ebay) [US]
North Castle Hardwoods [US]
The Winfield Collection [US]
These sites sell thin pieces of cherry lumber (and other types of wood as well), which might be useful if you’d like to make your own plywood blocks.
[wood]

Woodlike Matsumura [JP]
Woodlike Matsumura (International website)
A one-stop shop for printmaking supplies in Japan. Based in Tokyo, but they now (as of Dec 2023) ship overseas.
[wood][block][knife][chisel][paper][baren][brush][ink][paint]

Baren Studio Kikuhide [JP]
The shop of legendary baren maker Gotou Hidehiko. Located in Japan, but they can ship overseas. Their site is in Japanese, but they have an English web page with information about their products. You can place an order via email (English seems to be ok) and then make the payment through PayPal.
[baren]

Goods Japan [JP]
Goods Japan sells a variety of products that are made in Japan or made for the Japan market. They sell carving knives, chisels, brushes, and other tools that are useful for mokuhanga. Located in Japan, so shipping overseas may be a bit costly.
[chisel][brush][baren]

Awagami Factory [JP]
Awagami makes and sells paper that is suitable for printmaking. Located in Japan, so shipping overseas may be expensive. You can find some of their papers sold by other retailers (e.g. Blick, Hiromi, Jackson’s, etc) to avoid some of the shipping cost.
[paper]

Kitaro [JP]
Kitaro makes and sells paper for printmaking. Be warned, however, that when I tried purchasing from them, they never fulfilled my order, and they never responded to my emails about it. Other printmakers have ordered from them successfully though, so your mileage may vary. Their paper seems great for mokuhanga though.
[paper]

Jackson’s [UK]
Located in England, Jackson’s sells general art supplies, including some for printmaking.
[wood][block][knife][chisel][paper][brush][ink][paint]

Intaglio Printmaker [UK]
Intaglio Printmaker specializes in printmaking supplies.
[wood][block][knife][chisel][baren][ink][paint][paper]

Polymetaal [EU]
Based in the Netherlands, Polymetaal designs and produces printmaking equipment, primarily for Western techniques such as intaglio and lithography; but they also sell tools and materials for mokuhanga.
[wood][block][knife][chisel][baren][ink][paint][paper]

Japanese Paper Company [EU]
Japanese Paper Company sells a variety of washi, some of which may be suitable for printmaking.
[paper]