Illiteracy is a spectrum, not an absolute. More importantly, conquistadors were usually members of the 'middling' class, including artisans, craftsmen, and professionals (like lawyers and scribes). Most conquistadors probably could read some and at least write their names. Even if they couldn't there would have been a scribe on the expedition in their professional capacity or as just another member. Conquest expeditions were business ventures as much as they were military ones. The members of the company had shares in the enterprise and had contributed varying amounts. Bookkeepers and scribes were essential to the basic running of the expedition.

As to the general question of accommodating an illiterate population, yes, there were professional scribes and official notaries that were paid to write for others. When it came to petitioning the crown, Spanish law recognized the importance of protecting the poor and destitute. Consequently, the legal system had its own bureaucrats tasked with representing the poor and any legal matters they might have or petitions to the bureaucracy/crown (sometimes called the abogado de pobres, procurador de pobres, defensor de pobres). In the Americas, a special position was created just for indigenous petitioners/litigants (procurador de indios). If one qualified for the service, the fees for producing and submitting documents were waived as were the legal/notarial fees.

For some more on these issues see

Seven Myths of the Spanish Conquest

The First Letter From New Spain

The Men of Cajamarca

Justice by Insurance: The General Indian Court of Colonial Mexico and the Legal Aides of the Half-real

Law and the Transformation of Aztec Culture, 1500-1700

Empire of Law and Indian Justice in Colonial Mexico