Meaning of "...on the week of ..."

"The week of May 10th" means the whole week including that date. Normally you would use the Monday (which May 10, 2021 is) to identify the week, although it's not compulsory and sometimes you'll see another day used. Normally if used in business or education it will refer to the Monday-Friday week (at least in countries with Saturday and Sunday off).

As to "We would like to know your availability on the week of May 10th from 10:00 - 14:00." I think you're right, and they want to know your availability between those times every day. It probably means they can only schedule an appointment/meeting between those times each day and they want to know what days are best for you.

• https://www.phrasemix.com/phrases/the-week-of-date
• https://forum.english.best/t/the-week-of-the-12th/30368