As far as we know to-day the question is still open.
No one has yet produced a finite p-group which can be proved not to occur as the group of Fp–rational points of a connected unipotent algebraic Fp–group, but neither has anybody shown that every p-group does occur. What is known can be summarised as follows.

1. All abelian p-groups occur.
   Choose, for every non–negative integer m, the truncated Witt–vector group Wm. It is a connected unipotent algebraic group over Fp of dimension m, and Wm(Fp) is cyclic of order pm. Finite direct products of the Wm give every abelian p-group.

2. All groups whose nilpotence class is < p occur.
   If P has class < p the Lazard correspondence identifies P with the group obtained from a finite–dimensional nilpotent Lie algebra L over Fp by the BCH formula (which is a finite polynomial because the class is < p). Choosing a basis of L realises P as U(Fp) where U=exp(L) is a connected unipotent algebraic group defined over Fp.
   In particular every p-group of order pn with n≤p satisfies class < p, so every such group is known to occur.

3. “Algebra groups’’ give many further examples.
   Whenever J is a finite–dimensional nilpotent associative Fp–algebra, the set 1+J with multiplication (1+x)(1+y)=1+x+y+xy is the Fp–points of the connected unipotent linear algebraic group 1+J. These algebra groups contain, for example, all unitriangular matrix groups UTn(Fp), hence all their quotients. On the other hand (Evseev, Eick–Leedham-Green, …) there are many p-groups which are not algebra groups, so this construction is far from exhaustive.

4. Necessary conditions that are always satisfied.
   • |U(Fp)| is a pure power pd where d=dim U, and |U(Fq)|=qd for every q=pr.
   • U(Fp) is nilpotent and its centre is non-trivial.
   These conditions are, however, met by every finite p-group, so they give no obstruction.

5. No known obstruction beyond those in (4).
   Various ideas one might try (powerful groups, exponent bounds, Engel identities, the structure of the lower central series, …) are all defeated by examples that do occur (unitriangular groups, Witt-vector groups, …).

6. There is no known uniform construction covering all p-groups.
   It is easy to embed any finite p-group P in some UTm(Fp); what is missing is a way of cutting it out by polynomial equations so that the resulting closed, connected, smooth subgroup has exactly P as its Fp–points and still stays unipotent over every extension field Fpr.

In short:

• The answer is “yes’’ for large and well understood classes (all abelian groups, all groups of class < p, all algebra groups, …).

• No counter-example is known.

• A general theorem covering every finite p-group, or a proof that such a theorem is impossible, is presently not available.

So the problem remains open.