Ambiguity function of multiple sines
Sum of sines isn't a concrete waveform, there is no general form of its ambiguity function. But for any given sum of sines you can calculate this function by evaluating cross correlation for various time and Doppler deviations. Though one rule exists, if the number of sinewaves in composition grows, the ambiguity function will become narrower in both frequency and time dimensions, since such a signal will be more and more noise like. 

Since autocorrelation isn't linear function of some signal $s$ (its quadratic as you know cause it is sum of multiplications of $s$ and its shifted copy), so you can't express autocorrelation and more generally ambiguity function of sines composition as a sum of their autocorrelations (or amb. func-s.). Actually it is the sum of entering sines' autocorrelation functions and all possible entering sines' cross correlation functions. It could be something huge. So the most general way is to compute of such a function for given sines sum directly with knowing formula of autocorrelation.