The word "emoji" is the romanization of two Japanese words (絵 and 文字) that together mean picture character.   Emojis essentially marry an ancient form of writing, pictography, to a modern technology, smart phones.  

It would seem that California Secretary of State will not accept an emoji as a business name.  Section 21000(a) of the Title 2 of the California Code of Regulations provides:

Business entity names must use the English alphabet or Arabic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) or symbols as listed in Section 21002(b) or a combination thereof. 

Although the Secretary of State cites numerous provisions of the Corporations Code as authority for this regulation (8, 110, 201, 2106, 2601, 5008, 5122, 7122, 9122, 12214, 12302, 13409, 15901.08, 15909.05, 17701.08 & 17708.0), none of these statutes actually prohibit the use of emojis.  In fact, the word "emoji" has yet to make an appearance in any California statute.   

The closest that any of these statutes comes to banning emojis is Section 8 which provides "Whenever any notice, report, statement, or record is required or authorized by this code, it shall be made in writing in the English language".   But the Secretary of State's regulation allows names to include symbols and emojis are symbols.   This begs the question of the Secretary of State's authority to discriminate against emojis.

The symbols listed in Section 21002(b) are: @, #, $, %, ^, &, *, +, =.  What, if anything, do these symbols have in common?  They all appear on the row above the QWERTY row on a United States keyboard.