The Douay-Rheims version is a translation of the Vulgate. The Vulgate to Psalms seems to have gone through multiple revisions. I looked through all the versions I could find easily and found these translations of the word in question:
So there seems to be no support from the Vulgate for translating "God of Jacob."
I looked at older versions of the Douay-Rheims Bible and I found that the earlier versions had "God of heaven," which would be an appropriate translation of Dei caeli.
The 1610 (editio princeps) of the Douay-Rheims Old Testament says:
He that dwelleth in the helpe of the Higheſt, shal abide in the protection of the God of heauen.
I see the same language in these 1610 and 1635 editions.
The language "God of Jacob" seems to have originated in the 18th century. I couldn't find the 1749-50 edition, which apparently underwent a revision by Challoner. But the 1790 edition says:
He, who dwelleth in the aid of the moſt High, ſhall abide under the protection of the God of Jacob.
The editions from 1844 and 1846 change the wording but keep the "God of Jacob" innovation. The 1858 version only changes the capitalization of one word. It seems that this was the time that the wording was fixed. The version available on the internet seems to come from an 1899 edition (as described here).
Interestingly, Haydock's version (1855) gets the "God of heaven" right again:
He that dwelleth in the aid of the Most High, shall abide under the protection of the God of Heaven.
In a footnote he notes that "of heaven" isn't found in the Hebrew. This version (1883), which chooses to copy an authorized edition from 1812 but also to abridge Haydock's commentary, incongruently keeps the comment on "of heaven" but doesn't have those words in the text itself.
Since the change from "God of heaven" to "God of Jacob" seems to have happened in the midst of a big revision of the Bible, I would imagine that the change was just accidental, probably because "God of Jacob" is more common in the psalms than "God of heaven." Since there seems to be no textual support, from the Vulgate or otherwise, for "God of Jacob," I think it was just a printing mistake in one version that was copied in every edition afterwards.