There are a couple of pipe & tobacco & cigar stores in Budapest (some of them also sell cigarettes, although most of the well-known cigarette brands are available at major kiosks, especially in shopping malls).
Budapest Shopping Map gives you a user-friendly guide on where you can buy tobacco, pipes, cigars in Budapest Hungary. The map icon I have used is the green flames, and you can find further details of the stores at the bottom:
I have made good use of the list of the Hungarian pipe club site (Pipa Klub: pipaclub.hu). Based on their description, I had the impression that the most serious pipe store in Budapest is Gallwitz pipe and pearl shop in Régi Posta utca 7-9. (see the map above), founded in 1880 by Leopold Gallwitz, then continued by his sons, the Gallwitz brothers. Besides selling pipes (e.g. Peterson, Vauen, Savinelli, Dunhill, Missouri Meerschaum, Charatan, GBD Chanell etc.) and contemporary as well as antique pipes (from 22,000 to 65,000 HUF), they also offer walking sticks on sale (and repair). The other major section is pearl jewelry.
Address: Régi Posta utca 7-9. Budapest 1052
Phone: 00-36-30-297-5000 or 00-36-1-318-5139
Cigar Shop in Mammut Shopping Mall
Although it’s easy to find the shopping mall, the pipe store is slightly hidden on the 3rd floor of Mammut II. building (the one on the right side). Their shelves are featuring e.g. Peterson, Brebbia, Vauen, Hillson, Savinelli, Falcon, Vauen churchwarden in addition to normal cigarettes, zippos, etc.
Pipatórium in the 8th district, Budapest
Specialized in pipes and pipers, they have a wide range of tobaccos, pipes, etc. and the salesmen are informed too.
Address: 1085 Budapest József krt. 38.
Opening hours: Mon-Fri 10:00-18:00
Tobacco Shop in Florian Shopping Center
Phone: 00-36-1- 250-0080 / extension: 149
Address: 1033 Budapest Flórián tér 6-9., the first floor of the center
Opening hours: Mon-Fri: 10:00-18:00, Sat: 09:00-13:00
Davidoff Tobacco Store in Vaci utca
Mid-range pipes from top to bottom. The vendor is a pipe-smoker himself, so he is well-informed.
Address: Vaci utca 13. Budapest (Vaci utca shopping street in district V.)
Pipe store in Sas utca
It’s close to Deák tér metro station (also to Arany János metro station). Pipes: from top to bottom brands, tobaccos: moderate range.
Address: 1051 Budapest Sas utca 5.
Phone: 00-36-1-266-5085 or 00-36-1-30-565-4652)
Opening hours: Mon-Thu 10-17h; Fri 10-16h