Is there an A value for the carbaldehyde and vinyl groups?
Comparing the axial conformations of vinylcyclohexane and
  cyclohexancarbaldehyde: Which molecule is more strained?


As you point out, "A-values" can be used to get some idea of the relative "size" of a substituent in the cyclohexane system.  More specifically, the A-value reflects both the steric size of the substituent and the length of the bond connecting the substituent to the cyclohexane ring (the longer the bond, the further away the substituent and steric interactions are therefore reduced).  

The A-value is just the free energy difference between the axial and equatorial conformers. Since destabilizing steric interactions can occur in the axial conformation, the larger the substituent, less axial and more equatorial conformer will be present.



(image source)

As a consequence, the "larger" the substituent the larger the A-value. The A-value for a methyl group is 1.8, using

$$\mathrm{\Delta G = -RT*ln(K)}$$

we can solve for K and find that the axial/equatorial ratio for methylcyclohexane is ~ 5/95 (as expected, the less sterically strained equatorial conformer predominates) at room temperature.



(image source)

Here is a link to a nice table of A-values.  The A-value for a vinyl group ($\ce{-HC=CH2}$) is 1.6, while it is only 0.7 for the aldehyde group ($\ce{-CHO}$). Hence the aldehyde group is the smaller of the two. Or said differently, axial vinyl-cyclohexane would be more strained than axial cyclohexane-carbaldehyde