If members of the UK electorate were to report issues with the UK electoral system, and parties, persons and/or agencies related to it, would the EU have a legal right to investigate and if so what would those powers of investigation extend to...?
Part of the treaties is the Charter of Fundamental Rights of the European Union, of which in Article 40 is
Every citizen of the Union has the right to vote and to stand as a candidate at municipal elections in the Member State in which he or she resides under the same conditions as nationals of that State.
But that protects only the rights of non-UK european citizens in UK elections, not those of UK citizens.
Article 2 of the treaties states
The Union is founded on the values of respect for human dignity, freedom, democracy, equality, the rule of law and respect for human rights, including the rights of persons belonging to minorities. These values are common to the Member States in a society in which pluralism, non-discrimination, tolerance, justice, solidarity and equality between women and men prevail.
This is enforced by the institutions and can lead to a process according to Article 7.
So non-UK european citizens could appeal to the Court of Justice and the european institutions could start a process after Article 7.