Is it possible to make a vacuum balloon filled with electromagnetic radiation float? Given the right materials and structure, could you theoretically use radiation pressure to keep a equilibrium amount of pressure inside a special vacuum balloon so that it would float. (ignoring the absurd amount of radiation needed for a typical party sized balloon).
 A: Electromagnetic radiation does exert pressure, but it's terrible at it. For a totally reflecting surface, the pressure is proportional to the intensity (the numbers will turn out to be so large that we're ignoring small factors from reflection, etc):
$$P = {I \over c}$$
substituting the MKS numbers for balancing 1 atmosphere of outside pressure:
$$10^5 \:\mathrm{N/m^2} = {I \over {3 \times 10^8\:\rm m/s}}$$
$$ I = 3 \times 10^{13} \: \rm W/m^2$$ 
Keeping even a $1\rm m^2$ party-sized balloon inflated against a standard atmosphere using light requires power many times the electrical power generation capacity of the entire United States (~1000 GW).
A: In light of the better answers I have seen here, I want to update my previous answer and add a few sources.
In general relativity the pressure and energy of an enclosed photon "gas" (box of light) are connected by $U=3PV$ where U is energy, P is pressure and V is the box volume.  Photons behave differently from particles that do have a rest mass, and the equivalent inertial mass is twice the expected value.  So, it is $m = 2U/c^2 = 6PV/c^2$.  Let us insert numbers, using $P=1\cdot10^5 N/m^2$, $V=1m^3$ and $c=3\cdot 10^8 m/s$.  We find the mass of the enclosed light to be $6.7\cdot 10^{-12}kg$.  Compared to the mass of $1m^3$ of air (1.22kg) this is light, so the balloon will indeed float.  
The energy stored in the light is 600kJ for the $1m^3$ balloon.  That is not outrageously large, but the photons will be hitting the inner surface at a rate of at least $f=c/2R\approx 250MHz$.  Note that $R=0.62m$ for a $1m^3$ balloon.  So, we have a power density of $600kJ\cdot 250\cdot 10^6 Hz = 1.5\cdot 10^{14} W/m^2$ hitting the inside of the balloon.  As others here have pointed out, the balloon will evaporate in a few nano-seconds.  But it will float.
Sources: Mass in GR, Komar mass, and 
Carlip's paper

Below is the previous answer I gave.

No.  A balloon floats because the mass of the enclosed matter is less than the mass of the displaced medium.
Wood floats because one cubic meter has a mass of 600kg while one cubic meter of water has a mass of 1000kg.
A cubic meter helium-filled balloon floats because at 1 atmosphere pressure its mass is only 14% of the same-size air balloon.  It would still float if it was filled to 5 times the ambient air pressure, because then its mass still would only be 70% of the equivalent air mass.
In other words, pressure is not the deciding factor.  If the balloon has less mass than the mass it displaces, it will float.
A: To answer the is question properly, you  must calculate how much light energy would  need to be contained in the balloon in order to to exert one atmosphere of pressure, and then calculate the equivalent mass (using $E = Mc^2$, of course).  You would divide that mass by the volume of the balloon to obtain the minimum equivalent mass density of the light field in the balloon.  
If the equivalent mass density of the light field is less than the mass density of the atmosphere, the net buoyant force is equal to the weight of the light field in the balloon, minus the weight of an equal volume of air.  As long as that force exceeds the weight of the balloon itself (its "empty weight"), the balloon will float.
Note that Bob Jacobsen's answer addresses the intensity of light at the inner surface of the balloon needed to exert 1 atmosphere of pressure, but does not address the total energy of the light field contained in the balloon. A very rough estimate of the amount of light energy would be the intensity (watts/square meter) times the diameter of the balloon divided by the speed of light, times the cross-sectional area of the balloon -- for which we can use $E = IV/c$  (Energy = intensity x volume/speed of light).
The mass equivalent of the light field energy in the balloon, is $M =E/(c^2) = IV/c^3$.  The mass density of air at sea level is approximately (1.225 kg/m^2, so a light-filled balloon can float if $M/V = I/c^3 < 1.225 kg/m^2$.  If we take Bob's calculation of the required intensity as correct, then the condition is $$3x10^{13}< 1.225x(3x10^8)^3$$, or $$3 x 10^{13} < 3.3 x 10^{25}$$.
The condition is obviously met, so the answer is yes, a light-filled balloon could float (but the reflectivity of its inner surface had better be on the order of $(1-10^{-12}$)!
A: One way to think about this is to compare the mass density of air at one atmosphere to the mass density of photons to generate that same pressure. This is like comparing air to helium gas to fill the balloon.
TLDR: Because photons move much faster than atoms, a photon filled balloon doesn't need as much energy&matter when filled with photons:  It'll float.  Might burn up, but it'll float.
In algebraic detail:
The pressure of light completely reflecting off a surface is $2I/c$, where $I$ is the intensity.  The energy density of light is also $I/c$.  And to get from energy density to mass density, you have to divide by $c^2$ (from $E = mc^2$). So we get:
$I = cP/2$
$\rho_E = I/c = P/2$
$\rho_m = \rho_e/c^2 = P/{2c^2}$
But for a close-to-ideal gas, like air or helium, $PV = NkT$ can also be written in terms of number density of molecules as $P = \rho_N kT$ or mass density using the molecular mass m as $P = \rho_m kT / m$.  We can use that to generate a similar formula for mass density for a matter (non-photon) gas:
$\rho_m = P m/kT$
Comparing those two, what we're really interested in is which mass density is smaller (if its photons, the balloon floats).  And that becomes in turn whether $2c^2$ is greater than $kT/m$, which is to say $c^2$ is greater than $(kT/2) / m$
But $kT/2$ is the typical kinetic energy (in one dimension) of thermal motion, so $(kT/2) / m$ is just $v^2$ for a molecule. And that's much, much less than for a photon.
Incidentally, this comparison works for air vs helium too.  Helium "molecules" are lighter than air "molecules", so move faster, so you need less contained mass to get the same pressure:  A helium-filled balloon floats.
