Can an Agent have a logon with a blank password?

No.

Example:
Define and run a job using the local Administrator account on a PC.  The local Administrator account on the PC has a “blank” password; i.e. – enter the user name only and do not key anything into the password field.  The logon is allowed on the PC.  However, running a job using this profile is not allowed.  The error message matches the failure reason so it is correct. 

Job started at 05/26/2011 10:39:17 CDT

Job Name  . . . . :  localadmin

Job ID  . . . . . :  1005

Job Description . :  ken

Run Number  . . . :  1035

Agent Name  . . . :  KALVERSONXP_NEWAGENT

Agent OS  . . . . :  Windows

Server Name . . . :  kalversonxp.helpsystems.com:7472 Schedule Enterprise Server

 Starting process with command: bin\Win32\RobotTermSocket.exe  'Administrator' kalversonxp 1306424357360 C:\WINDOWS\system32\cmd.exe

 Password:

Unable to login user

1327: Logon failure: user account restriction.  Possible reasons are blank passwords not allowed, logon hour restrictions, or a policy restriction has been enforced.

Process exited with value: 123 The process ended unexpectedly.

Solution:

This was an intentional change to the LogonUser() API by MicroSoft.  See the following:

3.2.2 Restrictions on the Use of Blank Passwords

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Users who do not password-protect their accounts can only log on at their physical computer console: the monitor, keyboard, and mouse that is physically connected to their computer. This restriction only applies to local user accounts, not to domain user accounts.

Caution: If your computer is not in a physically secured location, you should assign strong passwords to all local user accounts. Failure to

do so allows anyone with physical access to the computer to log on using a user account that does not have a password. This is especially

important for portable computers, which should always have strong passwords on all local user accounts.