No, at present male Gerim and their male children do not have an inheritance in the land of Israel. Inheritance of property comes through the male line, meaning father to son.
Male converts do not become a part of any of the twelve tribes. And thus, their male children do not have a portion in the land at this time. This was a case decided by the court of Moshe Rabbeinu and mentioned in the written Torah according to Rashi's commentary. (See VaYikra 24:10-12 for details.) Rather, they are part of the congregation or community of Israel (עדת ישראל, קהלת ישראל).
This actually places them in the status of representing the whole, (הצבור) and not the individual. Like others who represent the whole of the Jewish people, meaning Kohanim, they do not possess an inheritance in the land.
This concept is actually mentioned in Midrash HaGadol (Volume 8 of Mossad HaRav Kook edition, parshat Pinchas 26:54-56, pp. 459-460, see also Bava Bathra 122a, Ein Yaacov to Bava Bathra 8:15, Yalkut Shimoni on the Torah, parshat Pinchas 773:7, and Yalkut Shimoni to Nach 29:1) which says that in the future, the converts will become a 13th tribe. This is actually alluded to in Yechezkel's prophecy, chapter 48:18-19, concerning the structure of the third Temple. It refers to an unnamed group called "the city workers" (העבד העיר) who will be given a special piece of land in Israel that is contributed by all of the tribes together.
They are recognized as converts because of their title (העבדי העיר). The double 'Heh' is a sign of them actually being Gerim like is found in Pirkei Avot 5:21-22 in regard to 'Ben Bag Bag' and 'Ben Heh Heh'. They were both children of converts. It is no accident that mention of them follows after Avot 5:20 which concludes by saying, "May it be Your will, L-rd, our G-d and G-d of our fathers, that the Temple will be rebuilt speedily in our days, and our portion in Your Torah is given." Meaning that they don't have a portion up until that time.
This double 'Heh' sign alludes back to the 'Heh' that was added to the names of both Avraham and Sarah when they made the covenant with G-d. (See Bereshit 17:1-8, 15-16 for details.)
It is worth pointing out that this is to distinguish from female converts and female children from a male convert. When a woman marries, she takes on the tribal status of her husband. If she marries a man of a particular tribe, she and all her future children are members of that tribe and are thus allotted a portion in the land of that tribe.
