Yes. There is both a duty to disclose and permit inspection of certain records (e.g. those that adversely affect either party's case or support another party's case) and a right for the court to demand access to other specific records.
In england-and-wales the applicable rules are found in Part 31 of the Civil Procedure Rules. For example:
Standard disclosure
31.6 Standard disclosure requires a party to disclose only– (a) the documents on which he relies; and (b) the documents which – (i)
adversely affect his own case; (ii) adversely affect another party’s
case; or (iii) support another party’s case; and (c) the documents
which he is required to disclose by a relevant practice direction.
Specific disclosure
(1) The court may make an order for specific disclosure or specific
inspection.
(2) An order for specific disclosure is an order that a party must do
one or more of the following things – (a) disclose documents or
classes of documents specified in the order; (b) carry out a search to
the extent stated in the order; (c) disclose any documents located as
a result of that search.
Party's control
31.8 (1) A party’s duty to disclose documents is limited to documents which are or have been in his control.
(2) For this purpose a party has or has had a document in his control
if – (a) it is or was in his physical possession; (b) he has or has
had a right to possession of it; or (c) he has or has had a right to
inspect or take copies of it.
Right of inspection
31.3 (1) A party to whom a document has been disclosed has a right to inspect that document except where – (a) the document is no longer in
the control of the party who disclosed it; (b) the party disclosing
the document has a right or a duty to withhold inspection of it, or
(c) paragraph (2) applies.
(2) Where a party considers that it would be disproportionate to the
issues in the case to permit inspection of documents within a category
or class of document disclosed under rule 31.6(b) – (a) he is not
required to permit inspection of documents within that category or
class; but (b) he must state in his disclosure statement that
inspection of those documents will not be permitted on the grounds
that to do so would be disproportionate.
Inspection and copying
31.15 Where a party has a right to inspect a document– (a) that party must give the party who disclosed the document written notice of his
wish to inspect it; (b) the party who disclosed the document must
permit inspection not more than 7 days after the date on which he
received the notice; and (c) that party may request a copy of the
document and, if he also undertakes to pay reasonable copying costs,
the party who disclosed the document must supply him with a copy not
more than 7 days after the date on which he received the request.
And, in case there's any doubt that this applies to electronic records:
Meaning of document
31.4 In this Part – ‘document’ means anything in which information of any description is recorded;
