The period of Japanese history that is most often called the period of reunification is the period of the early Tokugawa shogunate.  During this time, Japan was under pressure from European countries that had reached Japan.  The most important of these was Portugal.  The Spanish and Dutch were also involved to some extent.  The main way in which Japan felt pressured was through the introduction of Christianity.  The Portuguese in particular wanted to bring missionaries along with their traders.  The Japanese wanted the trade, but the missionaries soon caused problems in Japan.  This was one of the major factors behind the closing of Japan during the "sakoku" period.

What internal pressures did Japan face during its reunification period?

When you talk about "reunification" I assume that you are talking about the time when Toyotomi Hideyoshi took control of Japan and gave rise to the Tokugawa shogunate.

I would say that the major internal pressure at this time came from the actions of all of the separate daimyo.   These feudal lords tried to keep power for themselves against the attempts of the center to become more dominant.  These attempts led to a period of war that was really quite harmful to Japan's society and economy.  So the internal pressure came from a desire to end this state of constant war on the one hand and the desire of the various daimyo to keep power on the other.