Yes, you can.
The service is called MNP (Mobile Number Portability). 
You can request the MNP reservation number from your current provider (Docomo, Au, Softbank retail shops, or online customer support). 

We need the MNP to have more than 8 days before it expires when you sign up with us.

Request MNP number from Docomo

Request MNP number from Au (In Japanese only)

Request MNP number from Softbank (In Japanese only)

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The sign-up process would be as follows.

1. Request an MNP number from your current provider

2. Go to our Sign-up page

3. Click on the checkbox below

4. Provide the required information

4. Submit and complete the rest of the application