Is it permissible to make up with a non-Jewish store owner before Shabbos that you will take items from his store on Shabbos and pay for them after Shabbos?
If not - is it rabbinic or biblical? May one use the items on Shabbos?
R Yosef Kushner answers this question (pp. 60ff in his book Commerce and Shabbos) and forbids it.
He first explains one can complete a necessary transaction on Shabbat from a Jew if three conditions are met: one does not use the term "buy" or "sell", one makes no reference to "paying" or "getting paid" after Shabbat and one does not refer to the item by its cost or weigh during the course of the sale. As such one could take an object (after having made a previous agreement) or one could say "I will make the necessary arrangements after Shabbat". He notes it is important that this method should only be used when it is necessary for Shabbat or a mitzva. (see this other answer for more details).
However he notes that
According to many poskim, even when the above three conditions are met, one should not purchase an item from a gentile store on Shabbat.
He quotes the Rema (OC 500:1) that even the permissible methods of purchasing on Shabbat should only be used when purchasing from a fellow Jew but not from a Gentile, and gives three reasons
(see the original at length for sources)
This would seem to be permitted - both the buying and the using - if:
(1) No discussion about money happens on Shabbos
(2) There's an Eruv that permits one to carry from the shop to one's house.
(3) The items were within the Techum of Shabbos (about 1 Km) before Shabbos started.
(4) The Remo in Shulchan Aruch סימן רנב adds that the items were not made on Shabbos specifically for this Jew.
See Kitzur Shulchan Aruch סימן עג - באיזה אפן מתר לתת קודם שבת מלאכה לאינו יהודי ולהשאיל ולהשכיר לו כלים in 73:9 where he writes:
וְאֵינוֹ יְהוּדִי שֶׁאֵינוֹ אֻמָּן, אֶלָּא שֶׁיֵּשׁ לוֹ חֲנוּת שֶׁמּוֹכֵר מִנְעָלִים וְכַדּוֹמֶה, מֻתָּר לְיִשְֹרָאֵל הַמַּכִּירוֹ לִקַּח מִמֶּנּוּ בַּשַׁבָּת וּלְנָעֳלָם, וּבִלְבַד שֶׁלֹּא יִקְצֹץ עִמּוֹ דְּמֵי הַמִּקָח, וְגַם לֹא יִהְיוּ דְּבָרִים שֶׁהוּבְאוּ מִחוּץ לַתְּחוּם