Missing birth data is the single most common obstacle to a reading, and it is usually solvable.
Birth time determines the Ascendant (rising sign) and the house structure, which is a large share of what makes a chart specific to you rather than to everyone born that week. Without it, some of the reading is still available and some genuinely is not.
Start with documents. Hospital records, birth certificates in some countries, baby books, immunisation cards and family letters all carry times more often than people expect. A parent remembering "just before the morning news" is more useful than nothing.
Where no record exists, astrologers use rectification: working backwards from well-dated life events to test which Ascendant (rising sign) and house placements fit. It is careful, slow work and it is never presented as certainty.
If you would rather not go through that, a reading based on your Moon and planetary positions alone is still substantive — it just describes your inner weather and timing more than the outer circumstances of your life.