Integer Wavelet with Only Positive coefficients
If you are talking about the wavelet by itself, no, because   it is a  zero-mean function (by construction), hence there are positive and negative values. 

For wavelet coefficients from data, no, because of linearity: if for some $x$, $W(x)$ is positive, then $W(-x)=-W(x)$ is negative.

Note that for some signal models, with certain multiscale schemes, you could achieve your goal. You can find hereafter two references:


Lossless data hiding based on integer wavelet transform



  In data hiding, pieces of information represented by the data are
  hidden in the cover media. In some applications, people do care about
  the cover media. That is, the hidden data and the cover media may be
  closely related. For this type of data embedding, in addition to
  perceptual transparency, for some applications such as medical
  diagnosis and law enforcement, it is desired to invert the marked
  media back to the original cover media after the hidden data have been
  retrieved. The marking techniques satisfying this requirement are
  referred to as lossless, distortion-free or invertible data hiding
  techniques. From this point of view, it is observed that most of the
  current digital watermarking algorithms are not lossless.



Distortionless data hiding based on integer wavelet transform



  A novel distortionless image data hiding algorithm based on integer
  wavelet transform that can invert the stego-image into the original
  image without any distortion after the hidden data are extracted is
  proposed. This algorithm hides data into one (or more) middle
  bit-plane(s) of the integer wavelet transform coefficients in the
  middle and high frequency subbands. It can embed much more data
  compared with the existing distortionless data hiding techniques and
  satisfy the imperceptibility requirement. The image histogram
  modification is used to prevent greyscales from possible overflowing.
  Experimental results have demonstrated the validity of the algorithm.