There are lots of reasons that an IFR certified aircraft may not be legal to fly in IMC. An airplane must have had an altimeter and pitot/static check within the last 24 months. If using VORs for navigation, a VOR check is required within 30 days. If you are planning on using GPS approaches or departures, then your database must be current.
Since you start off your checkride by filing and flying an IFR flight plan, then it you would need an airplane that is currently capable of IFR flight. However, the current PTS does not require that an actual ATC clearance be received:
III. AREA OF OPERATION: AIR TRAFFIC CONTROL CLEARANCES AND PROCEDURES
NOTE: The ATC clearance may be an actual or simulated ATC clearance
based upon the flight plan.
If your designated examiner agrees beforehand that you will use a simulated clearance, then you would not need an IFR certified airplane. Otherwise I believe you would.
From the FARs:
§91.205 Powered civil aircraft with standard category U.S. airworthiness certificates: Instrument and equipment requirements.
(d) Instrument flight rules.
(1) Instruments and equipment specified in paragraph (b) of this section, and, for night flight, instruments and equipment specified in paragraph (c) of this section.
§91.411 Altimeter system and altitude reporting equipment tests and inspections.
(a) No person may operate an airplane, or helicopter, in controlled airspace under IFR unless—
(1) Within the preceding 24 calendar months, each static pressure system, each altimeter instrument, and each automatic pressure altitude reporting system has been tested and inspected and found to comply with appendices E and F of part 43 of this chapter;
And from the Instrument Practical Test Standards (FAA-S-8081-4E PTS Instrument)
Aircraft and Equipment Required for the Practical Test
The instrument rating applicant is required by 14 CFR part 61 to provide an airworthy, certificated aircraft for use during the practical test.
Its operating limitations must not prohibit the TASKS required on the
practical test. Flight instruments are those required for controlling
the aircraft without outside references. The required radio equipment
is that which is necessary for communications with air traffic control
(ATC), and for the performance of two of the following nonprecision
approaches: very high frequency omnidirectional range (VOR),
nondirectional beacon (NDB), global positioning system (GPS) without
vertical guidance, localizer (LOC), localizer-type directional aid
(LDA), simplified directional facility (SDF), or area navigation
(RNAV) and one precision approach: instrument landing system (ILS),
GNSS landing system (GLS), localizer performance with vertical
guidance (LPV) or microwave landing system (MLS).
GPS equipment must be instrument flight rules (IFR) certified and contain the current database.
Training is a different issue since, except for your cross-countries, you do not need to file an IFR flight plan to practice approaches and navigation solely by reference to instruments.