Just wanted to post a few scriptures ahead to give context for an answer at the end.
Now then, you and all these people, get ready to cross the Jordan River into the land I am about to give to them—to the Israelites. 3I will give you every place where you set your foot, as I promised Moses. Joshua 1:2
tell them that the flow of the Jordan was cut off before the ark of the covenant of the Lord. When it crossed the Jordan, the waters of the Jordan were cut off. Joshua 4:7
23For the Lord your God dried up the Jordan before you until you had crossed over. The Lord your God did to the Jordan what he had done to the Red Sea b when he dried it up before us until we had crossed over. 24He did this so that all the peoples of the earth might know that the hand of the Lord is powerful and so that you might always fear the Lord your God.” Joshua 4:23
1Now when all the Amorite kings west of the Jordan and all the Canaanite kings along the coast heard how the Lord had dried up the Jordan before the Israelites until they a had crossed over, their hearts melted in fear and they no longer had the courage to face the Israelites. Joshua 5:1
2At that time the Lord said to Joshua, “Make flint knives and circumcise the Israelites again.” Joshua 5:2
4Now this is why he did so: All those who came out of Egypt—all the men of military age—died in the wilderness on the way after leaving Egypt. 5All the people that came out had been circumcised, but all the people born in the wilderness during the journey from Egypt had not. Joshua 5:5
They were still uncircumcised because they had not been circumcised on the way. 8And after the whole nation had been circumcised, they remained where they were in camp until they were healed.
9Then the Lord said to Joshua, “Today I have rolled away the reproach of Egypt from you.” So the place has been called Gilgal c to this day. Joshua 5:7
So now getting to your question;
"These were their days of vulnerability. Why did God choose to circumcise the Israelites after they had crossed the Jordon River into enemies' territories?"
Quite often we see God weakening people so His strength can be seen in human weakness to accomplish great feats and show forth Himself as we see what he did through the new generation of Israel at that time.
I would venture to guess that their circumcision was a sign of their faith in the Lord. They obeyed His voice and depended on the Lord's words to give them the land of their inheritance. Yes they would be weak in the land of their enemies until they healed and their trust in the Lord is a great sign of their faith in Him to protect them during this time.