Using the approximation
$$\Delta \mathrm{center}=\frac{m_p}{m_\odot}\cdot\frac{\mathrm{dist}_p}{r_\odot}$$
and data from List of gravitationally rounded objects of the Solar System - Wikipedia:
Name Distance Mass/[kg] Mass Distance ΔCenter
/km /kg /sun mass /sun radius
--------------------------------------------------------------
Sun 0.00E+000 1.99E+030 1.00E+000 0.00E+000 0.00000
Mercury 5.79E+007 3.30E+023 1.66E-007 8.32E+001 0.00001
Venus 1.08E+008 4.87E+024 2.45E-006 1.55E+002 0.00038
Earth 1.50E+008 5.97E+024 3.00E-006 2.15E+002 0.00065
Mars 2.28E+008 6.42E+023 3.23E-007 3.27E+002 0.00011
Jupiter 7.78E+008 1.90E+027 9.55E-004 1.12E+003 1.06735
Saturn 1.43E+009 5.69E+026 2.86E-004 2.05E+003 0.58576
Uranus 2.87E+009 8.68E+025 4.37E-005 4.12E+003 0.18007
Neptune 4.50E+009 1.02E+026 5.15E-005 6.46E+003 0.33278
Which sums to a shift of 2.17 sun radii if all planets were aligned. Jupiter clearly dominates (about half of the total effect) and would be enough to move the center-of-gravity outside the sun. Only the heavy outer planets (Jupiter, Saturn, Uranus, Neptune) have a non negligible influence.
C# source code for reproducibility:
void Main()
{
// http://en.wikipedia.org/wiki/List_of_gravitationally_rounded_objects_of_the_Solar_System
Body[] planets={
new Body{Name="Mercury",Distance=57909175,Mass=3.302E23},
new Body{Name="Venus",Distance=108208930,Mass=4.8690E24},
new Body{Name="Earth",Distance=149597890,Mass=5.9742E24},
new Body{Name="Mars",Distance=227936640,Mass=6.4191E23},
new Body{Name="Jupiter",Distance=778412010,Mass=1.8987E27},
new Body{Name="Saturn",Distance=1426725400,Mass=5.6851E26},
new Body{Name="Uranus",Distance=2870972200,Mass=8.6849E25},
new Body{Name="Neptune",Distance=4498252900,Mass=1.0244E26},
};
var bodies=new[]{Sun}.Concat(planets);
bodies.Select(planet=>new{
Name=planet.Name,
Distance=planet.Distance.ToString("E2"),
Mass=planet.Mass.ToString("E2"),
DistanceInSunRadii=planet.DistanceInSunRadii.ToString("E2"),
MassInSuns=planet.MassInSuns.ToString("E2"),
CenterOfGravityShift=planet.CenterOfGravityShift.ToString("n5")
}).Dump();
}
static Body Sun=new Body{Name="Sun",Distance=0,Mass=1.98855E30};
const double SunRadius=696342;
class Body
{
public string Name{get;set;}
public double Distance{get;set;}// in km
public double Mass{get;set;}// in kg
public double MassInSuns{get{return Mass/Sun.Mass;}}
public double DistanceInSunRadii{get{return Distance/SunRadius;}}
public double CenterOfGravityShift{get{return MassInSuns*DistanceInSunRadii;}}
}