The problem addressed in this paper concerns the frictional contact between an elastomer and a rigid body with randomly rough surfaces. The work is accomplished on the basis of two crucial elements: a framework for generating random geometries and a robust frictional contact algorithm, both of which are realised in the approach of Isogeometric analysis (IGA) for its high accuracy and robustness. For the former, a new Isogeometric framework for random geometry modelling is proposed, which combines the random field generation based on Karhunen-Loève expansion theory with Non-Uniform Rational B-Spline (NURBS) interpolation method. For the latter, a mortar-based frictional contact algorithm in 2D large deformation regime is adopted incorporating a modified closest point projection method for detection of contact. Numerical experiments are conducted with several settings such as 'rough-smooth', 'smooth-rough' and 'rough-rough' contact, depending on which side of the contact pair the randomly rough surface belongs to. The ratio of the global coefficient of friction to the prescribed local one and the ratio of true contact area to the nominal contact area are characterised under these settings, and factors like the root mean square roughness and correlation length of the random surface and the external traction are discovered to have a significant influence on the two ratios.