Is Saju Based on the Lunar Calendar or the Solar Calendar?

Neither. Saju is calculated on a basis that's neither lunar nor solar — the solar-term calendar, which measures time by the sun's position. Whether you remember your birthday as a lunar date or a solar one, the Ten-Thousand-Year calendar converts it to the same underlying moment, so just tell it what you know. And the chart that comes out of that conversion is, itself, not a fixed fate — just a reference coordinate.

Why are solar terms the basis?

Because both the year and the month in a Saju chart are set by solar terms. The year turns over not on January 1st of the solar calendar but at Ipchun (roughly February 4th), and the month turns over not on the first day of a lunar month but only once a solar term has passed. That's why someone born in January or in early February can find their solar-calendar year and their Saju year don't match — someone born on February 1st being read under last year's zodiac sign is a case of exactly that.

I only know my lunar birthday — is that okay?

That's fine. The Ten-Thousand-Year calendar is a table that converts both lunar and solar dates into Sixty Gapja stem-branch pairs, so whichever one you give, it lands on the same moment. Being born in a leap lunar month is no different — the Saju month is set by solar terms, so a chart for a leap-month birth is drawn up just as solidly. The stem-branch pairs drawn this way are the very characters that make up the Iljin and Day Pillar on the Sixty Gapja system.