First of all from the fact this halacha is sourced in the gemara (Berachot 8a), one sees there is no need to own a Chumash, since certainly in those times there was no personal Chumash to be had.
R Avihud Schwartz notes (here, 3a) that R Ovadia Yosef wrote
One who has a sefer Torah in his possession and who knows the
cantillation and vowels by heart should ideally read shenayim mikra
ve-echad targum from the sefer Torah, for it is very holy.
In the footnote there, R. Yosef mentions that this was indeed the
practice of the Ari. He would go to the synagogue or study hall, take
the sefer Torah out of the aron, and read the weekly parasha from it.
Clearly the reference is to one who has access to a sefer Torah, not just one who owns his personal sefer Torah as very few people do.