Can I synthesize an Amide simply by heating an amine and a carboxylic acid (with a vacuum pump)?
It can be done but it is not a good reaction. From Lundberg et al. [1]:

Formally, the amide bond is formed through the condensation of a carboxylic acid and an amine with the release of one equivalent of water. This reaction has been considered challenging due to the competing acid–base reaction, which occurs when the amine and the carboxylic acid are mixed. Although the amide bond can be formed from the corresponding ammonium carboxylate salt upon heating, this reaction has generally been considered to be of limited preparative value. Furthermore, the high activation barrier for the direct coupling of a carboxylic acid and an amine can only be overcome using forcing reaction conditions.

Use of borate ester catalysis under Dean–Stark conditions has recently been reported to give good yields without resorting to acyl activation groups [2].
References

Lundberg, H.; Tinnis, F.; Selander, N.; Adolfsson, H. Catalytic Amide Formation from Non-Activated Carboxylic Acids and Amines. Chem. Soc. Rev. 2014, 43 (8), 2714–2742. DOI: 10.1039/C3CS60345H. (PDF)
Sabatini, M. T.; Boulton, L. T.; Sheppard, T. D. Borate Esters: Simple Catalysts for the Sustainable Synthesis of Complex Amides. Sci. Adv. 2017, 3 (9), e1701028. DOI: 10.1126/sciadv.1701028.