What wire gauge is needed to replace old wires in a vintage chandelier?
If you're in the US, NEC likely applies. Article 402 covers fixture wires, and explains what types and sizes are allowed. 

Type

Table 402.3 lists the types of wires allowed to be used as fixture wires. If you're going to rewire the fixture, you'll have to use a type of wire listed in this table.


FFH-2
HF, HFF
KF-1, KF-2, KFF-1, KFF-2
PAF, PAFF
PF, PFF
PGF, PGFF
PTF, PTFF
RFH-1, RFH-2, RFHH-2, RFHH-3
SF-1, SF-2, SFF-1, SFF-2
TF, TFF
TFN, TFFN
XF, XFF
ZF
ZFF
ZHF


Size

Table 402.5 lists the allowable ampacity for different sizes of wire. You'll want to use a wire that's sized properly for the load you'll be serving.  In most cases, the maximum bulb size you'll have to account for is 100 Watt. Which means you'll likely only be looking at about 833 mA (100 Watts/120 Volts), so you can probably get away with 18 AWG wire.


  


402.6 says that fixture wires cannot be smaller than 18 AWG, so you'll have to use at least that size wire.

Overcurrent Protection

402.12 points to 240.5 for overcurrent protection of fixture wires, which tells you what size overcurrent is required for different sized wires. 240.5(B)(2) lists the size and length of fixture wires, when connected to circuits with various sizes of overcurrent protection.


  National Electrical Code 2014
  
  Article 240 Overcurrent Protection
  
  I. General
  
  240.5 Protection of Flexible Cords, Flexible Cables, and Fixture Wires.
  
  (B) Branch-Circuit Overcurrent Device.
  
  (2) Fixture Wire. Fixture wire shall be permitted to be
  tapped to the branch-circuit conductor of a branch circuit in
  accordance with the following:
  
  (1) 20-ampere circuits — 18 AWG, up to 15 m (50 ft) of run length
  (2) 20-ampere circuits — 16 AWG, up to 30 m (100 ft) of run length
  (3) 20-ampere circuits — 14 AWG and larger
  (4) 30-ampere circuits — 14 AWG and larger
  (5) 40-ampere circuits — 12 AWG and larger
  (6) 50-ampere circuits — 12 AWG and larger  


Conduit/Tube Fill

You may also have to refer to Table 1 of Chapter 9, to determine the maximum number of wires that can be in a conduit or tube.