Using the Dividend Discount Model when no current dividend exists? Say a company is doing an IPO and we expect them to start paying dividends in 5 years?
I understand how we could use the Dividend Discount Model to predict their stock price 4 years from the IPO. But what about the day of the IPO?  

How can we use the Dividend Discount Model to value this stock before it pays dividends?
With what you say and using the DDM we know that

$P_{t=4} = \frac{D_{t=5}(1+g)}{r-g}$

Via the above equation, it is assumed that from year $5$, dividends will annually grow at a $g$ rate and that investors' required rate will be $r$, both forever. Since this calculation is nothing but a simplification (future growth and return rates will a fortiori change, etc...), we will not do more complicated.

Taking into account the share of debt in the capital structure of the company, say $X\%$, we can compute the average cost of capital (average over equity and debt), $r'$, needed to discount $P_{t=4}$. One may have an idea of $X$ or make assumptions about it. It follows that $r'$ is such that

$r' = X\% r_D + (1-X\%) r$

where $r_D$ is the interest rate or the cost of debt (naturally $r_S &lt; r$). Thus the company's IPO may be valued by involving that 

$P_{t=0} = \frac{P_{t=4}}{(1+r')^4}$

Finally, as you can see there are many assumptions in this valuation, which is the job of experts, with the most fundamental about $D_{t=5}$.