The USSR invaded Poland on September 17th, 1939.
By September 19th, Lavrentiy Beria issued Order No. 0308, to "organize a Prisoner-of-War Administration under the NKVD USSR".
By the time this order was issued, roughly how many Polish soldiers had already surrendered to the USSR?
I'm not finding any references which include daily casualty numbers for the invasion of Poland, but I think it nevertheless something to note that the order to organize POW Administration wouldn't have been a reaction to how many prisoners they had so far, but rather in line with what they were projecting to have. In that regards, the Soviets had roughly 250,000 or so Polish soldiers in their custody at the end of combat operations, which compares to a tad over half a million that the Germans took.
Now, back to the problem of specific numbers, this is in large part due to how secretive the Soviets were about the entire operation, very little solid information about whichever filtered out of the Soviet sphere during the Cold War. Just going through the campaign chronicle of Williamson's book for the first few days, he only mentions POWs of the first two days once, and then, only a representative example from a single battalion.
By 16 September there were between 100,000 and 150,000 Polish troops positioned along the eastern frontier zones.
In the far north, not far from the Latvian frontier and in the small town of Dzisna, there were several sharp clashes between Soviet troops and KOP units, local police and volunteers. The ferocity of these encounters can be judged from the fact that one battalion in the KOP Ga¤bokie Regiment lost over 50 per cent of its numbers, while the rest were taken prisoner.
So it is likely that those that did engage on the front suffered high casualties and many prisoners, but on the whole, it does look like most units were not making stands, but fighting retreats, or otherwise continuing to move. He makes an example of Wilno where the garrison, within a day of the invasion was instructed to "to make for the border and cross into Lithuania." As such, it seems evident that post of the prisoners the Soviets took were after the 19th, but I can't put a percentage on it based on the sources available. There is a Polish language work by Ryszard Szawaowski, but I don't have it, nor do I think it has been translated, but it might have more in-depth statistics if you do read Polish.
See: Poland Betrayed by D. G. Williamson