K-4 needs Green card, or EAD to get SSN
They are not required to ask for a SSN to register for school classes.
Anyone asking for an SSN can be dealt with, by demanding they provide the privacy act statement as required by The Privacy Act of 1974 (5 U.S.C. 552a).
QUOTE
Sec. 7(a) (1) It shall be unlawful for any Federal, State or local government agency to deny to any individual any right, benefit, or privilege provided by law because of such individual's refusal to disclose his social security account number.
(2) the provisions of paragraph (1) of this subsection shall not apply with respect to--
(A) any disclosure which is required by Federal statute, or
(B ) any disclosure of a social security number to any Federal, State, or local agency maintaining a system of records in existence and operating before January 1, 1975, if such disclosure was required under statute or regulation adopted prior to such date to verify the identity of an individual.
(b ) Any Federal, State or local government agency which requests an individual to disclose his social security account number shall inform that individual whether that disclosure is mandatory or voluntary, by what statutory or other authority such number is solicited, and what uses will be made of it.
http://www.usdoj.gov/oip/privstat.htm