You can build a chart without a birth time, and a great many people do. What you cannot build is a chart with houses. The honest version of the exercise is to separate the placements that are stable across the whole day from the ones that swing wildly, then read only the stable part.
Most of the chart, in fact. The Sun's sign is fixed except on the one or two days a year it ingresses, so for all but a sliver of birthdays your Sun sign is certain. Mercury never moves more than about 2 degrees in a day, and Venus and Mars never approach that, so their signs are reliable unless the birth date sits directly on an ingress. Jupiter through Pluto move so slowly by comparison that their signs and their mutual aspects are effectively time-independent on a single day.
The Moon is reliable when it holds one sign for the full day, which is true most days. The aspects between the slow-moving bodies, which are often the ones people find most descriptive, survive completely.
Houses, entirely. The Ascendant and every one of the twelve cusps depends on the exact time and the birth latitude. The Midheaven depends on time and is also latitude-sensitive, though less dramatically. House rulership chains break down with them, since the ruler of your first house is whatever planet rules the rising sign.
The Moon's house position, any luminary aspect to the angles, and the Part of Fortune all go with the houses. So does the distinction between a planet in the twelfth and a planet in the first, which in practice is the difference between a placement that hides and one that shows.
Many astrologers build the chart for 12:00 noon local time when the birth time is unknown. The reasoning is that noon splits the possible error, so a placement is at most twelve hours off in either direction rather than twenty-four in one. It also gives you a determinate Ascendant and a full set of houses to look at, which is useful if you want to see how the chart is structured.
The caution is that noon is a placeholder, not an estimate. It does not raise the probability that your real Ascendant matches the noon Ascendant. Two people with the same birthday born at 6 AM and 6 PM have different rising signs entirely, and the noon chart sits between them. Use the noon chart to demonstrate house mechanics, or to compare against a rectified chart later. Do not use it to claim a rising sign.
Lead with the placements you actually have and say so. A reading built on Sun, Moon, Mercury, Venus, Mars, and the outer planets by sign is a legitimate reading. It just answers different questions than a full chart does. It speaks to temperament, drive, communication style, and generational placement rather than to the specific arenas of life that houses map.
When you want houses, you have three routes. Find the birth time, which solves it outright. Build a solar or noon chart and treat the houses as a scaffold rather than a claim. Or look into rectification, which works backward from known life events to propose a time. Each has a cost and none of them is a substitute for the real record.
Astrology has never passed controlled testing, and no part of this is presented as established fact. The meanings assigned to signs and houses are conventions that have accumulated over centuries of practice, and their internal logic is what makes them interesting. A reader can get real value from thinking with the symbols while being clear that nothing here is measured, tested, or predictive in a scientific sense. That honesty costs nothing and prevents the worst kind of overclaiming.