Aeon Jusco Shinagawa

A large child friendly discount department store with everything. Perhaps Japan's version of Target? The basement is a supermarket, bakery and a food court with few fast food restaurants (high-chairs available). The first floor sells alcohol, gardening goods, pharmacy items, cosmetics and bicycles. The 2nd floor is where you want to be with kids, it has mens, womens and kids clothing and accessories, toys, baby food, diapers, strollers, cots and cribs, amusement game centre, a kids gym which costs a few hundred yen to enter and a free kids toy area. The 3rd floor has electrical goods, kitchen items, small furniture, bedding, a very good, cheap tailor and books (including a huge Japanese kids selection and some kids toys to keep them occupied while you look around).
The 2nd floor also has ride in cars for the kids with trolleys attached, they are great to take down to the basement to do your grocery shopping with.

Location: 4-12-5 Higashi-Shinagawa Shinagawa-ku, Tokyo (see other stores below)

Phone: 03-5715-8300 (will work in most navi's)

Train: 5 minute walk from Aomono-yokocho station on the Keikyu Line which is 10 minutes from Shinagawa station.

Bus: Catch the pink Aeon bus from Shinagawa station. The timetable is here (Japanese only).

Car: Parking available.

Hours: Open 9am to 11pm, restaurants and food shops are open till midnight.

Website: http://www.aeon.jp/aeon/shinagawaseaside/

Baby facilities: Changing tables are available in the toilets on each floor. The second floor has a large changing and feeding room near the baby section. Trolleys with child seats are also available and can be used through the store. The food court in the basement has highchairs.

Crawling:

Walking:

2-5 years:

5-10 years: They might like some of the amusement games.

Cost: Free
Stroller Friendly: Yes
Baby Changing Area: Yes
Feeding Room: Yes
Child Friendly Restaurant: Yes
Rainy Weather Option: Yes
Party Venue: No

Other Aeon Tokyo stores are located below:

2-6-1 Tokumaru, Itabashi-ku 03-5398-3131
Tobu Nerima Station http://www.aeon.jp/aeon/itabashi/index.html 

3-9-19 Nishi Kasai, Edogawa-Ku 03-3675-5111 
Nishi kasai Station http://www.aeon.jp/aeon/kasai/ 

37-13 Kitaminamachi, Ota-Ku 03-3727-4000
Ontakesan Station http://www.aeon.jp/aeon/ontakesan/

1-9-10 Shinonome, Koto-Ku 03-6221-3400
Toyosu Station http://www.aeon.jp/aeon/shinonome/

3-32-7 Umejima, Adachi-Ku 03-3852-2121
Nishi Arai Station  http://www.aeon.jp/aeon/nishiarai/index.html

6-7-15 Minamisuna, Koto-Ku 03-5677-3500
Minamisuna machi station  http://www.aeon.jp/aeon/minamisuna/

Kid-O-Kid Bornelund

A Large playroom full of Bornelund toys and activities including a jumping castle and bouncy runway. You will also find dressing up clothes, kitchen set-up, a 0-2 babies play area, a train set, and a huge ball pool with climbing frames and slides.

Location: Kanagawa-ken, Kawasaki-shi, Kawasaki-ku, Nisshincho 1-11, Kawasaki Le Front Building 9th Floor. See image here.

Phone: 045-650-1231 (Will work in most navi's)

Train Station: Kawasaki station

Parking: Only 25min drive from Minato-ku, Tokyo. It is a major shopping area with lots of parking available in the area.

Hours: 10:30 - 19:00（18:00 last entry）

Website: http://www.bornelund.co.jp/asobi/kidokid.html (Japanese Only)

Baby facilities: Changing and feeding room available

Crawling: Specific area for crawling babies.

Walking:

2-5 years:

5-10 years:

Cost: Children from 6 months to 12 years are 600 yen per 30min, over 12 years are 300 yen per 30min.
Stroller Friendly: Can be left at the front
Baby Changing Area: Yes
Feeding Room: Yes
Child Friendly Restaurant: Yes
Rainy Weather Option: Yes
Party Venue: Yes

Other Kid-o-Kid locations are:

1-1-17 Nishi Arai Sakaecho, Adachi-ku Tokyo 03-5888-6081
3-4-31 Minamisuna, Koto-Ku Tokyo 03-5677-0550
1-1-2 Utsukushigaoka, Aoba-ku Yokohama 045-482-4800 
Lalaport Tokyo Bay 2-1-1 Hama-cho, Funabashi, Chiba-ken 047-421-7790
4-6-5 Minatomarai, Yokohama 045-650-1231 

See the website (Japanese only) with the full list here.