Hello Nico and a warm welcome to you!
First things first - in Estonia, majority of countryside roads (at least those of "last mile" type) are gravel roads and some do not tend to be in good/satisfactory condition. If it happens to be a rainy summer, one might find enough adventure on those.
I'm sorry, I do not know your van's capabilities. Does it have diff locks, what not? Winch, hi-lift?
Good news is that in Aegviidu, there is an old artillery range (abandoned) on which one can test his vehicle (talking about softroading on real dirt roads, not some serious type of offroading). About 50 km from Tallinn (capital of Estonia) and publicily accessible.
If you have a Garmin GPS unit, good or even excellent maps are available (topographic maps combined with street/road navigation as well). Also, quite good printed atlas is available at every Statoil gas station (this one is called "Regio Atlas"). This Aegviidu area (not the range itself, but close to that) also offers quite a lot free-to-use camping sites, some even with lakes etc.
What I would suggest personally is to navigate through the outer perimeter of mainland (i.e. skip the islands), taking the smallest roads possible. I've done it on '84 Ford Escort, doable, wouldn't do it with modern 3-series BMW though (or comparable).
Of course, all that depends on your schedule... about a week should be planned for a trip not overrushed.
Also, North-East part of Estonia ("Ida-Virumaa" in local language) would be an experience - mining areas, power plants, adventurous roads and some forest deep enough that most of "normal europeans" may not feel comfortable.
That can be fit into the around-the-mainland trip as well, not exceeding the 1-week-limit.
That Aegviidu artilley range is also navigable with those GPS maps (in terms of dirt roads) so do not fear getting lost. If your plans take some more firm shape and dates will be fixed, I'm sure you can find some company from this site as well for a good safari!
If you plan some serious offroading/dirtroading alone, print out some phone numbers of those with capable rigs and willing to help. Just in case. Respective thread in this forum is http://forum.4x4.ee/viewtopic.php?f=5&t=4960
, in Estonian unfortunatley and I wouldn't rely on Google Translate ... But, perhaps some english-speaking dudes chime in here as well and give you some phone numbers to call when in trouble. Depending on your exact plans, that is.
Do let us know about your plans and sure we can come up with more detailed suggestions!